Logarithm - 01

$If log 10 a = p, log 10 b = q, then what is log 10 (a p b q) equal to ?$ $\eqalign{ & {\text{If}}\,{\log _{10}}a = p,\,\,\,{\log _{10}}b = q, \cr & {\text{then}}\,{\text{what}}\,\,{\text{is}}\,\,{\log _{10}}\left( {{a^p}{b^q}} \right)\,\,{\text{equal}}\,\,{\text{to}}? \cr}$

1p² + q²
2p² - q²
3p²q²
4^p²/_q²

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Answer:- 1

Explanation:-

Solution:

Given, log 10 a = p, log 10 b = q log 10 (a p b q) = log 10 a p + log 10 b q = p log 10 a + q log 10 b = p 2 + q 2

$\eqalign{ & {\text{Given}}, \cr & {\log _{10}}a = p,\,{\log _{10}}b = q \cr & {\log _{10}}\left( {{a^p}{b^q}} \right) = {\log _{10}}{a^p} + {\log _{10}}{b^q} \cr & = p{\log _{10}}a + q{\log _{10}}b \cr & = {p^2} + {q^2} \cr}$

$If log 3 log (3 x - 2) and log (3 x + 4) are in arithmetic progression, then x is equal to -$ $\eqalign{ & {\text{If}}\,\log 3\log \left( {{3^x} - 2} \right)\,{\text{and}}\,\log \left( {{3^x} + 4} \right)\, \cr & {\text{are}}\,{\text{in}}\,{\text{arithmetic}}\,\,{\text{progression}}\,{\text{,}} \cr & {\text{then}}\,x\,{\text{is}}\,{\text{equal}}\,{\text{to}} - \cr}$

1⁸/₃
2log 3⁸
3log 2³
48

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Answer:- 1

Explanation:-

Solution:

In arithmetic progression common ratio are equal tolog(3x−2)−log3=log(3x+4)−log(3x−2)=log(3x−2)log3=log(3x+4)log(3x−2)(∴loga−logb=logab)=log3xlog2log3=xlog3log4log2xlog3=xlog3log2log3=xlog3log4log2xlog2=xlog2=log4log2⇒x=log4log2log2⇒x=log8⇒x=log23In arithmetic progression common ratio are equal tolog(3x−2)−log3=log(3x+4)−log(3x−2)=log(3x−2)log3=log(3x+4)log(3x−2)(∴loga−logb=logab)=log3xlog2log3=xlog3log4log2xlog3=xlog3log2log3=xlog3log4log2xlog2=xlog2=log4log2⇒x=log4log2log2⇒x=log8⇒x=log23<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>In arithmetic progression common ratio are equal to</mtext></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>log</mi><mo>⁡</mo><mn>3</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mo>∴</mo><mi>log</mi><mo>⁡</mo><mi>a</mi><mo>−</mo><mi>log</mi><mo>⁡</mo><mi>b</mi><mo>=</mo><mi>log</mi><mo>⁡</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mi>x</mi></msup></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mi>log</mi><mo>⁡</mo><mn>3</mn><mi>log</mi><mo>⁡</mo><mn>4</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mi>log</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mi>log</mi><mo>⁡</mo><mn>3</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mi>log</mi><mo>⁡</mo><mn>3</mn><mi>log</mi><mo>⁡</mo><mn>4</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mi>log</mi><mo>⁡</mo><mn>4</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>x</mi><mo>=</mo><mi>log</mi><mo>⁡</mo><mn>4</mn><mi>log</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>x</mi><mo>=</mo><mi>log</mi><mo>⁡</mo><mn>8</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>x</mi><mo>=</mo><mi>log</mi><mo>⁡</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>2</mn><mn>3</mn></msup></mrow></mtd></mtr></mtable></math>

$\eqalign{ & {\text{In arithmetic progression common ratio are equal to}} \cr & \log \left( {{3^x} - 2} \right) - \log 3 \cr & = \log \left( {{3^x} + 4} \right) - \log \left( {{3^x} - 2} \right) \cr & = \frac{{\log \left( {{3^x} - 2} \right)}}{{\log 3}} \cr & = \frac{{\log \left( {{3^x} + 4} \right)}}{{\log \left( {{3^x} - 2} \right)}}\left( {\therefore \log a - \log b = \log \frac{a}{b}} \right) \cr & = \frac{{\log {3^x}}}{{\log 2\log 3}} \cr & = \frac{{x\log 3\log 4\log 2}}{{x\log 3}} \cr & = \frac{{x\log 3}}{{\log 2\log 3}} \cr & = \frac{{x\log 3\log 4\log 2}}{{x\log 2}} \cr & = \frac{x}{{\log 2}} \cr & = \log 4\log 2 \cr & \Rightarrow x = \log 4\log \,2\log 2 \cr & \Rightarrow x = \log 8 \cr & \Rightarrow x = \log {2^3} \cr}$

$The number of digits in 49 \times 517, when expressed in usual form, is -$ $\eqalign{ & {\text{The number of digits in }}{{\text{4}}^9} \times {{\text{5}}^{17}}{\text{,}} \cr & {\text{when expressed in usual form, is - }} \cr}$

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Answer:- 1

Explanation:-

Solution:

log (49 \times 517) = log (49) + log (517) = log (22) 9 + log (517) = log (218) + log (517) = 18 log 2 + 17 log 5 = 18 log 2 + 17 (log 10 - log 2) = 18 log 2 + 17 log 10 - 17 log 2 = log 2 + 17 log 10 = 0.3010 + 17 \times 1 = 17.3010 ∴ characteristic = 17, Hence, the number of digits in (49 \times 517) = 18

$\eqalign{ & \log \left( {{4^9} \times {5^{17}}} \right) \cr & = \log \left( {{4^9}} \right) + \log \left( {{5^{17}}} \right) \cr & = \log \left( {{2^2}} \right)9 + \log \left( {{5^{17}}} \right) \cr & = \log \left( {{2^{18}}} \right) + \log \left( {{5^{17}}} \right) \cr & = 18\log 2 + 17\log 5 \cr & = 18\log 2 + 17\left( {\log 10 - \log 2} \right) \cr & = 18\log 2 + 17\log 10 - 17\log 2 \cr & = \log 2 + 17\log 10 \cr & = 0.3010 + 17 \times 1 = 17.3010 \cr & \therefore {\text{characteristic}} = 17, \cr & {\text{Hence, the number of digits in }}\left( {{4^9} \times {5^{17}}} \right) = 18 \cr}$

$If log 2 = 0.30103, the number of digits in 450 is -$ $\eqalign{ & {\text{If}}\log 2 = 0.30103, \cr & {\text{the}}\,{\text{number}}\,{\text{of}}\,{\text{digits}}\,{\text{in}}\,{4^{50}}{\text{is}} - \cr}$

130
231
3100
4200

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Answer:- 1

Explanation:-

Solution:

log 40 = 50 log 4 = 50 log 22 = (50 \times 2) log 2 = 100 \times log 2 = (100 \times 0.30103) = 30.103 ∴ characteristic = 30, Hence, the number of digits in 450 = 31

$\eqalign{ & \log {4^0} \cr & = 50\log 4 \cr & = 50\log {2^2} \cr & = \left( {50 \times 2} \right)\log 2 \cr & = 100 \times \log 2 \cr & = \left( {100 \times 0.30103} \right) \cr & = 30.103 \cr & \therefore {\text{characteristic}} = 30, \cr & {\text{Hence, the number of digits in }}{{\text{4}}^{50}} = 31 \cr}$

$If log 2 = 0.3010 a n d log 3 = 0.4771, t h e v a l u e o f log 5 512 = ?$ $\eqalign{ & {\text{If}}\,\log 2 = 0.3010\,and\,\log 3 = 0.4771, \cr & the\,value\,of\,{\log _5}512 = ? \cr}$

12.870
22.967
33.876
43.912

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Answer:- 1

Explanation:-

Solution:

log5512=log512log5=log29log(102)=9log2log10−log2=(9×0.3010)1−0.3010=2.7090.699=2709699=3.876log5512=log512log5=log29log(102)=9log2log10−log2=(9×0.3010)1−0.3010=2.7090.699=2709699=3.876<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>5</mn></msub></mrow><mn>512</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>512</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>5</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>2</mn><mn>9</mn></msup></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>9</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>log</mi><mo>⁡</mo><mn>10</mn><mo>−</mo><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mn>9</mn><mo>×</mo><mn>0.3010</mn></mrow><mo>)</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mo>−</mo><mn>0.3010</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>2.709</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0.699</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>2709</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>699</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mn>3.876</mn></mtd></mtr></mtable></math>

$\eqalign{ & {\log _5}512 \cr & = \frac{{\log 512}}{{\log 5}} \cr & = \frac{{\log {2^9}}}{{\log \left( {\frac{{10}}{2}} \right)}} \cr & = \frac{{9\log 2}}{{\log 10 - \log 2}} \cr & = \frac{{\left( {9 \times 0.3010} \right)}}{{1 - 0.3010}} \cr & = \frac{{2.709}}{{0.699}} \cr & = \frac{{2709}}{{699}} \cr & = 3.876 \cr}$

$If a = lo g 8 225 and b = lo g 2 15, then a in terms of b is -$ $\eqalign{ & {\text{If}}\,{\text{a = lo}}{{\text{g}}_{\text{8}}}{\text{225}}\,{\text{and}}\,{\text{b = lo}}{{\text{g}}_{\text{2}}}{\text{15}}, \cr & {\text{then}}\,a\,{\text{in}}\,{\text{terms}}\,{\text{of}}\,b\,{\text{is}} - \cr}$

1^b/₂
2^2b/₃
3b
4^3b/₂

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Answer:- 1

Explanation:-

Solution:

a=log8225=log23(152)=23log215=2b3a=log8225=log23(152)=23log215=2b3<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mi>a</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>8</mn></msub></mrow><mn>225</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>2</mn><mn>3</mn></msup></mrow></mrow></msub></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>2</mn></msub></mrow><mn>15</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>b</mi></mrow><mn>3</mn></mfrac></mtd></mtr></mtable></math>

$\eqalign{ & a = {\log _8}225 \cr & = {\log _{{2^3}}}\left( {{{15}^2}} \right) \cr & = \frac{2}{3}{\log _2}15 \cr & = \frac{{2b}}{3} \cr}$

$If log 7 log 5 (x + 5 - - - - - \sqrt + x - - \sqrt) = 0, what is the value of x ?$ $\eqalign{ & {\text{If}}\,\,{\log _7}{\log _5}\left( {\sqrt {x + 5} + \sqrt x } \right) = 0, \cr & {\text{what}}\,\,{\text{is}}\,\,{\text{the}}\,\,{\text{value}}\,\,{\text{of}}\,\,x? \cr}$

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Answer:- 1

Explanation:-

Solution:

log 7 log 5 (x + 5 - - - - - \sqrt + x - - \sqrt) = 0 \Rightarrow log 5 (x + 5 - - - - - \sqrt + x - - \sqrt) = 70 = 1 \Rightarrow x + 5 - - - - - \sqrt + x - - \sqrt = 51 = 5 \Rightarrow (x + 5 - - - - - \sqrt + x - - \sqrt) 2 = 25 \Rightarrow (x + 5) + x + 2 x + 5 - - - - - \sqrt x - - \sqrt = 25 \Rightarrow 2 x + 2 x - - \sqrt x + 5 - - - - - \sqrt = 20 \Rightarrow x - - \sqrt x + 5 - - - - - \sqrt = 10 \Rightarrow x (x + 5) = (10 - x) 2 \Rightarrow x 2 + 5 x = 100 + x 2 - 20 x \Rightarrow 25 x = 100 \Rightarrow x = 4

$\eqalign{ & \,{\log _7}{\log _5}\left( {\sqrt {x + 5} + \sqrt x } \right) = 0 \cr & \Rightarrow {\log _5}\left( {\sqrt {x + 5} + \sqrt x } \right) = {7^0} = 1 \cr & \Rightarrow \sqrt {x + 5} + \sqrt x = {5^1} = 5 \cr & \Rightarrow {\left( {\sqrt {x + 5} + \sqrt x } \right)^2} = 25 \cr & \Rightarrow \left( {x + 5} \right) + x + 2\sqrt {x + 5} \sqrt x = 25 \cr & \Rightarrow 2x + 2\sqrt x \sqrt {x + 5} = 20 \cr & \Rightarrow \sqrt x \sqrt {x + 5} = 10 \cr & \Rightarrow x\left( {x + 5} \right) = {\left( {10 - x} \right)^2} \cr & \Rightarrow {x^2} + 5x = 100 + {x^2} - 20x \cr & \Rightarrow 25x = 100 \cr & \Rightarrow x = 4 \cr}$

$If log 3 x + log 9 - x 2 + log 27 x 3 = 9, then x equals -$ $\eqalign{ & {\text{If}}\,{\log _3}x + {\log _{\underline 9 }}{x^2} + {\log _{27}}{x^3} = 9, \cr & {\text{then}}\,\,x\,\,{\text{equals - }} \cr}$

13
29
327
4None of these

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Answer:- 1

Explanation:-

Solution:

log3x+log9–x2+log27x3=9⇒log3x+log32x2+log33x3=9⇒log3x+22log3x+33log3x=9⇒3log3x=9⇒log3x=3⇒x=33=27log3x+log9_x2+log27x3=9⇒log3x+log32x2+log33x3=9⇒log3x+22log3x+33log3x=9⇒3log3x=9⇒log3x=3⇒x=33=27<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><munder><mn>9</mn><mo>_</mo></munder></mrow></msub></mrow><mrow class="MJX-TeXAtom-ORD"><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>27</mn></mrow></msub></mrow><mrow class="MJX-TeXAtom-ORD"><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>32</mn></mrow></msub></mrow><mrow class="MJX-TeXAtom-ORD"><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>33</mn></mrow></msub></mrow><mrow class="MJX-TeXAtom-ORD"><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mn>3</mn></msub></mrow><mi>x</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>x</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mn>3</mn></msup></mrow><mo>=</mo><mn>27</mn></mtd></mtr></mtable></math>

$\eqalign{ & {\log _3}x + {\log _{\underline 9 }}{x^2} + {\log _{27}}{x^3} = 9 \cr & \Rightarrow {\log _3}x + {\log _{32}}{x^2} + {\log _{33}}{x^3} = 9 \cr & \Rightarrow {\log _3}x + \frac{2}{2}{\log _3}x + \frac{3}{3}{\log _3}x = 9 \cr & \Rightarrow 3{\log _3}x = 9 \cr & \Rightarrow {\log _3}x = 3 \cr & \Rightarrow x = {3^3} = 27 \cr}$

$If log x - 5 log 3 = - 2, then x equals -$ $\eqalign{ & {\text{If}}\,\,\log x - 5\log 3 = - 2, \cr & {\text{then }}x{\text{ equals - }} \cr}$

10.8
20.81
31.25
42.43

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Answer:- 1

Explanation:-

Solution:

logx−5log3=−2⇒logx−log35=−2⇒log(x35)=−2⇒x243=10−2=1100⇒x=243100=2.43logx−5log3=−2⇒logx−log35=−2⇒log(x35)=−2⇒x243=10−2=1100⇒x=243100=2.43<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mi>log</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mn>5</mn><mi>log</mi><mo>⁡</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>log</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mi>log</mi><mo>⁡</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mn>5</mn></msup></mrow><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>log</mi><mo>⁡</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mn>3</mn><mn>5</mn></msup></mrow></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mfrac><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>243</mn></mrow></mfrac><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn></mrow></msup></mrow><mo>=</mo><mfrac><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mn>100</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">⇒</mo><mi>x</mi><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>243</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>100</mn></mrow></mfrac><mo>=</mo><mn>2.43</mn></mtd></mtr></mtable></math>

$\eqalign{ & \log x - 5\log 3 = - 2 \cr & \Rightarrow \log x - \log {3^5} = - 2 \cr & \Rightarrow \log \left( {\frac{x}{{{3^5}}}} \right) = - 2 \cr & \Rightarrow \frac{x}{{243}} = {10^{ - 2}} = \frac{1}{{100}} \cr & \Rightarrow x = \frac{{243}}{{100}} = 2.43 \cr}$

$Iflog107=a,thenlog10(170)isequalto−Iflog107=a,thenlog10(170)isequalto−<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>If</mtext></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mn>7</mn><mo>=</mo><mi>a</mi><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mtext>then</mtext></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mn>70</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mtext>is</mtext></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mtext>equal</mtext></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mtext>to</mtext></mrow><mo>−</mo></mtd></mtr></mtable></math>$ $\eqalign{ & {\text{If}}\,\,{\log _{10}}7 = a,{\text{then}}\,\,{\log _{10}}\left( {\frac{1}{{70}}} \right)\,\, \cr & {\text{is}}\,\,{\text{equal}}\,\,{\text{to}} - \cr}$

1-(1 + a)
2(1 + a)^-1
3^a/₁₀
4¹/_10a

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Answer:- 1

Explanation:-

Solution:

log10(170)=log101−log1070=−log10(7×10)=−(log107+log1010)=−(a+1)log10(170)=log101−log1070=−log10(7×10)=−(log107+log1010)=−(a+1)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mn>70</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mn>1</mn><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mn>70</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>log</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mn>7</mn><mo>×</mo><mn>10</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mrow class="MJX-TeXAtom-ORD"><mi>log</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mn>7</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msub><mrow class="MJX-TeXAtom-ORD"><mi>log</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></mrow><mn>10</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math>

$\eqalign{ & {\log _{10}}\left( {\frac{1}{{70}}} \right) \cr & = {\log _{10}}1 - {\log _{10}}70 \cr & = - {\log _{10}}\left( {7 \times 10} \right) \cr & = - \left( {{{\log }_{10}}7 + {{\log }_{10}}10} \right) \cr & = - \left( {a + 1} \right) \cr}$

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Logarithm - 01

$If log 10 a = p, log 10 b = q, then what is log 10 (a p b q) equal to ?$ $\eqalign{ & {\text{If}}\,{\log _{10}}a = p,\,\,\,{\log _{10}}b = q, \cr & {\text{then}}\,{\text{what}}\,\,{\text{is}}\,\,{\log _{10}}\left( {{a^p}{b^q}} \right)\,\,{\text{equal}}\,\,{\text{to}}? \cr}$

$The number of digits in 49 \times 517, when expressed in usual form, is -$ $\eqalign{ & {\text{The number of digits in }}{{\text{4}}^9} \times {{\text{5}}^{17}}{\text{,}} \cr & {\text{when expressed in usual form, is - }} \cr}$

$If log 2 = 0.30103, the number of digits in 450 is -$ $\eqalign{ & {\text{If}}\log 2 = 0.30103, \cr & {\text{the}}\,{\text{number}}\,{\text{of}}\,{\text{digits}}\,{\text{in}}\,{4^{50}}{\text{is}} - \cr}$

$If log 2 = 0.3010 a n d log 3 = 0.4771, t h e v a l u e o f log 5 512 = ?$ $\eqalign{ & {\text{If}}\,\log 2 = 0.3010\,and\,\log 3 = 0.4771, \cr & the\,value\,of\,{\log _5}512 = ? \cr}$

$If a = lo g 8 225 and b = lo g 2 15, then a in terms of b is -$ $\eqalign{ & {\text{If}}\,{\text{a = lo}}{{\text{g}}_{\text{8}}}{\text{225}}\,{\text{and}}\,{\text{b = lo}}{{\text{g}}_{\text{2}}}{\text{15}}, \cr & {\text{then}}\,a\,{\text{in}}\,{\text{terms}}\,{\text{of}}\,b\,{\text{is}} - \cr}$

$If log 7 log 5 (x + 5 - - - - - \sqrt + x - - \sqrt) = 0, what is the value of x ?$ $\eqalign{ & {\text{If}}\,\,{\log _7}{\log _5}\left( {\sqrt {x + 5} + \sqrt x } \right) = 0, \cr & {\text{what}}\,\,{\text{is}}\,\,{\text{the}}\,\,{\text{value}}\,\,{\text{of}}\,\,x? \cr}$

$If log 3 x + log 9 - x 2 + log 27 x 3 = 9, then x equals -$ $\eqalign{ & {\text{If}}\,{\log _3}x + {\log _{\underline 9 }}{x^2} + {\log _{27}}{x^3} = 9, \cr & {\text{then}}\,\,x\,\,{\text{equals - }} \cr}$

$If log x - 5 log 3 = - 2, then x equals -$ $\eqalign{ & {\text{If}}\,\,\log x - 5\log 3 = - 2, \cr & {\text{then }}x{\text{ equals - }} \cr}$

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Logarithm - 01

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