Study91

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

116
217
318
419

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} \log (4^{9} \times 5^{17}) \\ = \log (4^{9}) + \log (5^{17}) \\ = \log (2^{2}) 9 + \log (5^{17}) \\ = \log (2^{18}) + \log (5^{17}) \\ = 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 (4^{9} \times 5^{17}) = 18 \end{aligned}

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

$\begin{aligned} The value of \\ \frac{6 \log_{10} 1000}{3 \log_{10} 100} is equal to - \end{aligned}$ $\eqalign{ & {\text{The value of }} \cr & \,\frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}100}}\,\,{\text{is equal to - }} \cr}$

10
21
32
43

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} \frac{6 \log_{10} 1000}{3 \log_{10}} \\ = \frac{6 \log_{10} 10^{3}}{3 \log_{10} 10^{2}} \\ = \frac{6 \times 3 \log_{10} 10}{3 \times 2 \log_{10} 10} \\ = \frac{18}{6} \\ = 3. \end{aligned}

$\eqalign{ & \frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}}} \cr & = \frac{{6{\text{ }}{{\log }_{10}}{{10}^3}}}{{3{\text{ }}{{\log }_{10}}{{10}^2}}} \cr & = \frac{{6 \times 3{\text{ }}{{\log }_{10}}10}}{{3 \times 2{\text{ }}{{\log }_{10}}10}} \cr & = \frac{{18}}{6} \cr & = 3. \cr}$

$\frac{\log \sqrt{8}}{\log 8} is equal to = ?$ $\frac{{\log \sqrt 8 }}{{\log 8}}\,\,{\text{is equal to = ?}}$

1¹/_√8
2¹/₄
3¹/₂
4¹/₈

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} \frac{\log \sqrt{8}}{\log 8} \\ = \frac{\log {(8)}^{\frac{1}{2}}}{\log 8} \\ = \frac{\frac{1}{2} \log 8}{\log 8} \\ = \frac{1}{2} \end{aligned}

$\eqalign{ & \frac{{\log \sqrt 8 }}{{\log 8}} \cr & = \frac{{\log {{\left( 8 \right)}^{\frac{1}{2}}}}}{{\log 8}} \cr & = \frac{{\frac{1}{2}\log 8}}{{\log 8}} \cr & = \frac{1}{2} \cr}$

$\begin{aligned} If \log_{1000} x = - \frac{1}{4}, \\ then the value of x is = ? \end{aligned}$ $\eqalign{ & {\text{If }}{\log _{1000}}x = - \frac{1}{4}{\text{, }} \cr & {\text{then the value of }}x{\text{ is = ? }} \cr}$

1¹/₁₀
2¹/₁₀₀
3¹/₁₀₀₀
4¹/₁₀₀₀₀

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} let \log_{1000} x = - \frac{1}{4} \\ \Rightarrow x = {(10000)}^{- \frac{1}{4}} \\ = {(10^{4})}^{- \frac{1}{4}} \\ = 10^{- 1} \\ = \frac{1}{10} \end{aligned}

$\eqalign{ & {\text{let }}{\log _{1000}}{\text{x}} = - \frac{1}{4} \cr & \Rightarrow {\text{ x}} = {\left( {10000} \right)^{ - \frac{1}{4}}} \cr & = {\left( {{{10}^4}} \right)^{ - \frac{1}{4}}} \cr & = {10^{ - 1}} \cr & = \frac{1}{{10}} \cr}$

$If lo g_{8} p = 25 and lo g_{2} q = 5, then -$ ${\text{If }}\,\,{\text{lo}}{{\text{g}}_8}{\text{p}} = 25\,\,{\text{and lo}}{{\text{g}}_2}{\text{q}} = 5\,,{\text{then - }}$

1p = q¹⁵
2p² = q³
3p = q⁵
4p³ = q

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} lo g_{8} p = 25 and lo g_{2} q = 5 \\ \Rightarrow p  = 8^{25} and q = 2^{5} \\ \Rightarrow p  = {(2^{3})}^{25} and q  = 2^{5} \\ \Rightarrow p  = 2^{75} and q  = 2^{5} \\ \Rightarrow p  = {(2^{5})}^{15} and q  = 2^{5} \\ \Rightarrow p  = q^{15} \end{aligned}

$\eqalign{ & {\text{lo}}{{\text{g}}_8}{\text{p}} = 25\,\,{\text{and lo}}{{\text{g}}_2}{\text{q}} = 5\, \cr & \Rightarrow {\text{p = }}{{\text{8}}^{25}}\,{\text{and q}} = {2^5} \cr & \Rightarrow {\text{p = }}{\left( {{2^3}} \right)^{25}}{\text{ and q = }}{2^5} \cr & \Rightarrow {\text{p = }}{{\text{2}}^{75}}{\text{ and q = }}{2^5} \cr & \Rightarrow {\text{p = }}{\left( {{2^5}} \right)^{15}}{\text{ and q = }}{2^5} \cr & \Rightarrow {\text{ p = }}{{\text{q}}^{15}} \cr}$

$\begin{aligned} The value of \\ lo g_{10} (0.0001) is = ? \end{aligned}$ $\eqalign{ & {\text{The value of }} \cr & {\text{lo}}{{\text{g}}_{10}}\left( {0.0001} \right)\,\,{\text{is }} = ? \cr}$

1¹/₄
2-¹/₄
3-4
44

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} lo g_{10} (0.0001) \\ = lo g_{10} (\frac{1}{1000}) \\ = lo g_{10} (\frac{1}{10^{4}}) \\ = lo g_{10} 10^{- 4} \\ = - 4 lo g_{10} 10 \\ = - 4 \end{aligned}

$\eqalign{ & {\text{lo}}{{\text{g}}_{10}}\left( {0.0001} \right)\, \cr & = {\text{lo}}{{\text{g}}_{10}}\left( {\frac{1}{{1000}}} \right) \cr & \, = {\text{lo}}{{\text{g}}_{10}}\left( {\frac{1}{{{{10}^4}}}} \right) \cr & \, = {\text{lo}}{{\text{g}}_{10}}{10^{ - 4}} \cr & \, = - 4\,{\text{lo}}{{\text{g}}_{10}}10 \cr & = - 4{\text{ }} \cr}$

$\begin{aligned} Determine the value of \\ lo g_{3 \sqrt{2}} (\frac{1}{18}) is = ? \end{aligned}$ $\eqalign{ & {\text{Determine the value of }} \cr & {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{18}}} \right)\,{\text{ is = ?}} \cr}$

12
2-2
3√2
4√3

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} lo g_{3 \sqrt{2}} (\frac{1}{18}) \\ = lo g_{3 \sqrt{2}} (\frac{1}{{(3 \sqrt{2})}^{2}}) \\ = lo g_{3 \sqrt{2}} {(3 \sqrt{2})}^{- 2} \\ = (- 2) lo g_{3 \sqrt{2}} 3 \sqrt{2} \\ = - 2 \end{aligned}

$\eqalign{ & {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{18}}} \right)\, \cr & = {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{{{\left( {3\sqrt 2 } \right)}^2}}}} \right)\, \cr & = {\text{lo}}{{\text{g}}_{3\sqrt 2 }}{\text{ }}{\left( {3\sqrt 2 } \right)^{ - 2}} \cr & = \left( { - 2} \right){\text{lo}}{{\text{g}}_{3\sqrt 2 }}3\sqrt 2 \cr & = - 2 \cr}$

$\begin{aligned} The value of \\ \log_{5} \frac{(125) (625)}{25} \\ is equal to - \end{aligned}$ $\eqalign{ & {\text{The value of}} \cr & {\log _5}\frac{{\left( {125} \right)\left( {625} \right)}}{{25}} \cr & {\text{is equal to - }} \cr}$

1725
25
33125
46

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} lo g_{5} \frac{(125) (625)}{25} \\ = lo g_{5} (\frac{5^{3} \times 5^{4}}{5^{2}}) \\ = lo g_{5} 5^{5} \\ = 5lo g_{5} 5 \\ = 5 \end{aligned}

$\eqalign{ & {\text{lo}}{{\text{g}}_5}\frac{{\left( {125} \right)\left( {625} \right)}}{{25}} \cr & = {\text{lo}}{{\text{g}}_5}\left( {\frac{{{5^3} \times {5^4}}}{{{5^2}}}} \right) \cr & = {\text{lo}}{{\text{g}}_5}{5^5} \cr & {\text{ = 5lo}}{{\text{g}}_5}5 \cr & = 5 \cr}$

09. The value of log₂ 16 is:

1¹/₈
24
38
416

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

Let log₂ 16 = n.
Then, 2ⁿ = 16 = 2⁴ ⇒ n = 4.
∴ log₂ 16 = 4.

10. If a^x = b^y, then:

1log ^a/_b = ^x/_y
2^{log a}/_{log b} = ^x/_y
3^{log a}/_{log b} = ^y/_x
4None of these

View Answer Discuss in forum Workspace Report

Answer:- 1

Explanation:-

Solution:

\begin{aligned} a^{x} = b^{y} \\ \Rightarrow \log a^{x} = \log b^{y} \\ \Rightarrow x \log a = y \log b \\ \Rightarrow \frac{\log a}{\log b} = \frac{y}{x} \end{aligned}

$\eqalign{ & {a^x} = {b^y} \cr & \Rightarrow \log {a^x} = \log {b^y} \cr & \Rightarrow x\log a = y\log b \cr & \Rightarrow {{\log a} \over {\log b}} = {y \over x} \cr}$

Hello Guest

Logarithm - 02

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

$\begin{aligned} The value of \\ \frac{6 \log_{10} 1000}{3 \log_{10} 100} is equal to - \end{aligned}$ $\eqalign{ & {\text{The value of }} \cr & \,\frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}100}}\,\,{\text{is equal to - }} \cr}$

$\frac{\log \sqrt{8}}{\log 8} is equal to = ?$ $\frac{{\log \sqrt 8 }}{{\log 8}}\,\,{\text{is equal to = ?}}$

$\begin{aligned} If \log_{1000} x = - \frac{1}{4}, \\ then the value of x is = ? \end{aligned}$ $\eqalign{ & {\text{If }}{\log _{1000}}x = - \frac{1}{4}{\text{, }} \cr & {\text{then the value of }}x{\text{ is = ? }} \cr}$

$If lo g_{8} p = 25 and lo g_{2} q = 5, then -$ ${\text{If }}\,\,{\text{lo}}{{\text{g}}_8}{\text{p}} = 25\,\,{\text{and lo}}{{\text{g}}_2}{\text{q}} = 5\,,{\text{then - }}$

$\begin{aligned} The value of \\ lo g_{10} (0.0001) is = ? \end{aligned}$ $\eqalign{ & {\text{The value of }} \cr & {\text{lo}}{{\text{g}}_{10}}\left( {0.0001} \right)\,\,{\text{is }} = ? \cr}$

$\begin{aligned} Determine the value of \\ lo g_{3 \sqrt{2}} (\frac{1}{18}) is = ? \end{aligned}$ $\eqalign{ & {\text{Determine the value of }} \cr & {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{18}}} \right)\,{\text{ is = ?}} \cr}$

$\begin{aligned} The value of \\ \log_{5} \frac{(125) (625)}{25} \\ is equal to - \end{aligned}$ $\eqalign{ & {\text{The value of}} \cr & {\log _5}\frac{{\left( {125} \right)\left( {625} \right)}}{{25}} \cr & {\text{is equal to - }} \cr}$

09. The value of log₂ 16 is:

10. If a^x = b^y, then:

Subject Wise

Exam Wise

NCERT
6 to 12

Paid Course

Practice Batch

Daily Current Affairs

Monthly Current Affairs

Topic Wise Current Affairs

State Wise Current Affairs

Prelims Facts

Hello Guest

Logarithm - 02

01. The number of digits in 49×517,when expressed in usual form, is - The number of digits in 49×517,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}

02. The value of 6 log1010003 log10100is equal to - The value of 6 log1010003 log10100is equal to - \eqalign{ & {\text{The value of }} \cr & \,\frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}100}}\,\,{\text{is equal to - }} \cr}

03. log8–√log8is equal to = ?log⁡8log⁡8is equal to = ?\frac{{\log \sqrt 8 }}{{\log 8}}\,\,{\text{is equal to = ?}}

04. If log1000x=−14, then the value of x is = ? If log1000x=−14, then the value of x is = ? \eqalign{ & {\text{If }}{\log _{1000}}x = - \frac{1}{4}{\text{, }} \cr & {\text{then the value of }}x{\text{ is = ? }} \cr}

05. If log8p=25and log2q=5,then - If log8p=25and log2q=5,then - {\text{If }}\,\,{\text{lo}}{{\text{g}}_8}{\text{p}} = 25\,\,{\text{and lo}}{{\text{g}}_2}{\text{q}} = 5\,,{\text{then - }}

06. The value of log10(0.0001)is =?The value of log10(0.0001)is =?\eqalign{ & {\text{The value of }} \cr & {\text{lo}}{{\text{g}}_{10}}\left( {0.0001} \right)\,\,{\text{is }} = ? \cr}

07. Determine the value of log32√(118) is = ?Determine the value of log32(118) is = ?\eqalign{ & {\text{Determine the value of }} \cr & {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{18}}} \right)\,{\text{ is = ?}} \cr}

08. The value oflog5(125)(625)25is equal to - The value oflog5(125)(625)25is equal to - \eqalign{ & {\text{The value of}} \cr & {\log _5}\frac{{\left( {125} \right)\left( {625} \right)}}{{25}} \cr & {\text{is equal to - }} \cr}

09. The value of log2 16 is:

10. If ax = by, then:

Subject Wise

Exam Wise

NCERT 6 to 12

Paid Course

Practice Batch

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

$\begin{aligned} The value of \\ \frac{6 \log_{10} 1000}{3 \log_{10} 100} is equal to - \end{aligned}$ $\eqalign{ & {\text{The value of }} \cr & \,\frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}100}}\,\,{\text{is equal to - }} \cr}$

$\frac{\log \sqrt{8}}{\log 8} is equal to = ?$ $\frac{{\log \sqrt 8 }}{{\log 8}}\,\,{\text{is equal to = ?}}$

$\begin{aligned} If \log_{1000} x = - \frac{1}{4}, \\ then the value of x is = ? \end{aligned}$ $\eqalign{ & {\text{If }}{\log _{1000}}x = - \frac{1}{4}{\text{, }} \cr & {\text{then the value of }}x{\text{ is = ? }} \cr}$

$If lo g_{8} p = 25 and lo g_{2} q = 5, then -$ ${\text{If }}\,\,{\text{lo}}{{\text{g}}_8}{\text{p}} = 25\,\,{\text{and lo}}{{\text{g}}_2}{\text{q}} = 5\,,{\text{then - }}$

$\begin{aligned} The value of \\ lo g_{10} (0.0001) is = ? \end{aligned}$ $\eqalign{ & {\text{The value of }} \cr & {\text{lo}}{{\text{g}}_{10}}\left( {0.0001} \right)\,\,{\text{is }} = ? \cr}$

$\begin{aligned} Determine the value of \\ lo g_{3 \sqrt{2}} (\frac{1}{18}) is = ? \end{aligned}$ $\eqalign{ & {\text{Determine the value of }} \cr & {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{18}}} \right)\,{\text{ is = ?}} \cr}$

$\begin{aligned} The value of \\ \log_{5} \frac{(125) (625)}{25} \\ is equal to - \end{aligned}$ $\eqalign{ & {\text{The value of}} \cr & {\log _5}\frac{{\left( {125} \right)\left( {625} \right)}}{{25}} \cr & {\text{is equal to - }} \cr}$

09. The value of log₂ 16 is:

10. If a^x = b^y, then:

NCERT
6 to 12