Question 12 Marks
Find $x,$ if : $log_x(5x - 6) = 2$
Answer$log_x(5x - 6) = 2$
$\Rightarrow 5x - 6 = x^2 ...[$ Removing logarithm $]$
$\Rightarrow x^2 - 5x + 6 = 0$
$\Rightarrow x^2 - 3x - 2x + 6 = 0$
$\Rightarrow x( x - 3 ) - 2( x - 3 ) = 0$
$\Rightarrow ( x - 2 )( x - 3 ) = 0$
$\therefore x = 2, 3.$
View full question & answer→Question 22 Marks
Find $x,$ if : $log_x 625 = - 4$
Answer$\log _x 625=-4$
$\Rightarrow 625=x^{-4} \ldots[$ Removing logarithm $]$
$\Rightarrow 5^4=\left(\frac{1}{x}\right)^4$
$\Rightarrow 5=\frac{1}{x} ....[$ Powers are same$,$ bases are equal $]$
$\Rightarrow x=\frac{1}{5}$
View full question & answer→Question 32 Marks
Evaluate : $log_38 \div log_916$
Answer$\log _3 8 \div \log _9 16$
$ \Rightarrow \frac{\log _3 8}{\log _9 16}$
$ \Rightarrow \frac{\log _{10} 8}{\log _{10} 3} \times \frac{\log _{10} 9}{\log _{10} 16}$
$ \Rightarrow \frac{3 \log _{10} 2}{\log _{10} 3} \times \frac{2 \log _{10} 3}{4 \log _{10} 2}$
$ \Rightarrow \frac{3}{2}$
View full question & answer→Question 42 Marks
Evaluate: $\log_ba \times \log_c b \times \log_a c.$
Answer$ \log _{\mathrm{b}} \mathrm{a} \times \log _{\mathrm{c}} \mathrm{b} \times \log _{\mathrm{a}} \mathrm{c} $
$ \Rightarrow \frac{\log _{10} a}{\log _{10} b} \times \frac{\log _{10} b}{\log _{10} c} \times \frac{\log _{10} c}{\log _{10} a}$
$ \Rightarrow 1$
View full question & answer→Question 52 Marks
Given $\log _{10} \mathrm{x}=2 \mathrm{a}$ and $\log _{10} \mathrm{y}=\frac{b}{2}.$ Write $10^{2 \mathrm{~b}+1}$ in terms of $\mathrm{y}$.
Answer$ \log _{10} y=\frac{b}{2}$
$ \Rightarrow y=10^{b / 2}$
$ \Rightarrow y^4=10^{2 b}$
$ \Rightarrow 10 y^4=10^{2 b} \times 10$
$ \Rightarrow 10^{2 b+1}$
$=10 y^4$
View full question & answer→Question 62 Marks
Given $\log _{10} \mathrm{x}=2 \mathrm{a}$ and $\log _{10} \mathrm{y}=\frac{b}{2}.$ Write $10^{\mathrm{a}}$ in terms of $\mathrm{x}.$
Answer$\log_{10}x = 2a$
$\Rightarrow x = 10^{2a} ...[$ Removing logarithm from both sides $]$
$\Rightarrow x^{1/2} = 10^a$
$\Rightarrow 10^a = x^{1/2}$
View full question & answer→Question 72 Marks
Simplify :$\log (a)^3÷ \log a$
Answer$\log (a)^3 \div \log a $
$=3 \log a \div \log a $
$=\frac{3 \log a}{\log a} $
$=3$
View full question & answer→Question 82 Marks
Simplify : $\log (a)^3- \log a$
Answer$\log (a)^3 - \log a$
$= 3 \log a - \log a$
$= 2 \log a$
View full question & answer→Question 92 Marks
If $log_5x = y,$ find $ 5^{2y+ 3}$ in terms of $x$.
Answer$\log_5 x = y ...[$ given $]$
$\Rightarrow 5^y = x$
$\Rightarrow (5^y)^2 = x^2$
$\Rightarrow 5^{2y} = x^2$
$\Rightarrow 5^{2y} x 5^3 = x^2 x 5^3$
$\Rightarrow 5^{2y + 3} = 125x^2$
View full question & answer→Question 102 Marks
If $log_{10}a = b,$ find $10^{3b - 2}$ in terms of a.
Answer$ \log _{10} \mathrm{a}=\mathrm{b}$
$ \Rightarrow 10^{\mathrm{b}}=\mathrm{a}$
$ \Rightarrow\left(10^{\mathrm{b}}\right)^3=\mathrm{a}^3 \ldots[\text { Cubing both sides ] }$
$ \Rightarrow \frac{10^{3 b}}{10^2}=\frac{a^3}{10^2} \ldots\left[\text { dividing both sides by } 10^2\right]$
$ \Rightarrow 10^{3 \mathrm{~b}-2}=\frac{a^3}{100}$
View full question & answer→Question 112 Marks
Solve for $x : \log_{10}(x - 10) = 1$
Answer$\log_{10} ( x - 10 ) = 1$
$\Rightarrow \log_{10} ( x - 10 ) = \log_{10}10$
$\Rightarrow x - 10 = 10$
$\Rightarrow x = 10 + 10$
$\Rightarrow x$ $= 20.$
View full question & answer→Question 122 Marks
Express $ \log_{10}2 + 1$ in the form of $ \log_{10}x .$
Answer$\log_{10}2 + 1= \log_{10}2 + \log_{10}10 ....[ \because \log_{10}10 = 1 ]$
$= \log_{10}2 \times 10 ....[ \log_am + \log_an = \log_amn ]$
$= \log_{10}20.$
View full question & answer→Question 132 Marks
Express the following in a form free from logarithm:$a \log x - b \log y = 2 \log 3$
AnswerConsider the given equation
$\operatorname{a \log} x-b \log y=2 \log 3$
$ \Rightarrow \log x^a-\log y^b=\log 3^2$
$ \Rightarrow \log \left(\frac{x^a}{y^b}\right)=\log 9$
$ \Rightarrow \frac{x^a}{y^b}=9$
$ \Rightarrow x^a=9 y^b$
View full question & answer→Question 142 Marks
Express the following in a form free from logarithm:$2 \log x + 3 \log y = \log a$
AnswerConsider the given equation
$2 \log x + 3 \log y = \log a$
$\Rightarrow \log x^2 + \log y^3 = \log a$
$\Rightarrow \log x^2y^3 = \log a$
$\Rightarrow x^2y^3 = a$
View full question & answer→Question 152 Marks
Express the following in a form free from logarithm :$2 \log x - \log y = 1$
AnswerConsider the given equation
$2 \log x-\log y=1$
$ \Rightarrow \log x^2 -\log y=1$
$ \Rightarrow \log \left(\frac{x^2}{y}\right)=\log 10$
$ \Rightarrow \frac{x^2}{y}=10$
$ \Rightarrow x^2=10 y$
View full question & answer→Question 162 Marks
Express in terms of $\log 2$ and $\log 3:\log 144$
Answer$\log 144= \log( 2 \times 2 \times 2 \times 2 \times 3 \times 3 )$
$= \log( 2^4 \times 3^2 )$
$= \log( 2^4 ) + \log( 3^2 ) ....[ \log_amn = \log_am + \log_an]$
$= 4 \log2 + 2 \log3 ....[ \log_am^n = n \log_am ]$
View full question & answer→Question 172 Marks
Express in terms of $\log 2$ and $\log 3$ : $\log 36$
Answer$\log 36 = \log( 2 \times 2 \times 3 \times 3 )$
$= \log( 2^2 \times 3^2 )$
$= \log( 2^2 ) + \log( 3^2 ) ....[ \log_amn = \log_am + \log_an ]$
$= 2\log2 + 2\log3 .....[ \log_am^n = n\log_am ]$
View full question & answer→Question 182 Marks
If $\log_{10}2 = a$ and $\log_{10}3 = b;$ express each of the following in terms of $'a\ '$ and $'b\ '$ :$ \log 60$
Answer$\log 60$
$= \log_{10}10 \times 2 \times 3 ...[ \log_amn = \log_am + \log_an ]$
$= 1 + \log_{10}2 + \log_{10}3 ...[ \because \log1010 = 1 ]$
$= 1 + a + b ...[ \because \log_{10}2 = a$ and $\log_{10}3 = b ]$
View full question & answer→Question 192 Marks
Given $\log_5y = n$. Express $5^{3n + 2}$ in terms of $ y.$
Answer$\log _5 y=n$
$5^n=y$
Consider $5^n=y$
$\Rightarrow\left(5^n\right)^3=y^3 $
$\Rightarrow 5^{3 n}=y^3 $
$\Rightarrow 5^{3 n} \times 5^2=y^3 \times 5^2 $
$\Rightarrow 5^{3 n+2}=25 y^3$
View full question & answer→Question 202 Marks
Given $\log_2x = m$. Express $2^{m - 3}$ in terms of $x.$
Answer$ \log _2 {x}=\mathrm{m}$
$ \Rightarrow 2^{\mathrm{m}}={x}$
$ \text { Consider } 2^{\mathrm{m}}={x}$
$ \Rightarrow \frac{2^m}{2^3}=\frac{x}{2^3}$
$ \Rightarrow 2^{m-3}=\frac{x}{8}$
View full question & answer→Question 212 Marks
If $\log _a m=n,$ express $a^{n-1}$ in terms of $a$ and $m.$
Answer$\log _a m=n$
$\Rightarrow a^n=m$
$ \Rightarrow \frac{a^n}{a}=\frac{m}{a}$
$ \Rightarrow a^{n-1}=\frac{m}{a}$
View full question & answer→Question 222 Marks
Evaluate : $\log_5125$
AnswerLet $\log_5 125 = x$
$\Rightarrow 5^x = 125$
$\Rightarrow 5^x = 5 \times 5 \times 5$
$\Rightarrow 5^x = 5^3$
$\Rightarrow x = 3$
Thus, $\log_5125 = 3$
View full question & answer→Question 232 Marks
Evaluate : $\log_51$
AnswerLet $\log_5 1 = x$
$\Rightarrow 5^x = 1$
$\Rightarrow 5^x = 5^0$
$\Rightarrow x = 0$
Thus, $\log_5 1 = 0$
View full question & answer→Question 242 Marks
Evaluate : $\log_{10}0.01$a
Answer$ \text { Let } \log _{10} 0.01=x $
$ \Rightarrow 10^x=0.01 $
$ \Rightarrow 10^x=\frac{1}{100} $
$ \Rightarrow 10^x=\frac{1}{10 \times 10} $
$ \Rightarrow 10^x=\frac{1}{10^2} $
$ \Rightarrow 10^x=10^{-2} $
$ \Rightarrow x=-2$
Thus, $\log _{10} 0.01=-2$
View full question & answer→Question 252 Marks
Find $x,$ if : $\log_5(x - 7) = 1$
AnswerConsider the equation
$\log_5(x - 7) = 1$
$\Rightarrow 5^1 = x - 7$
$\Rightarrow 5 = x - 7$
$\Rightarrow x = 5 + 7$
$\Rightarrow x = 12$
View full question & answer→Question 262 Marks
Find $x,$ if : $\log_x 2 = - 1.$
AnswerConsider the equation
$\log _x 2=-1$
$ \Rightarrow x^{-1}=2$
$\Rightarrow \frac{1}{x}=2$
$\Rightarrow x=\frac{1}{2}$
View full question & answer→Question 272 Marks
Find $x,$ if : $\log_3x = 0$
AnswerConsider the equation
$\log_3x = 0$
$\Rightarrow 3^0 = x$
$\Rightarrow 1 = \times$ or $\times = 1$
View full question & answer→Question 282 Marks
Find the logarithm of : $0.1$ to the base $10$
Answer$ \text { Let } \log _{10} 0.1={x}$
$ \therefore 10^{x}=0.1$
$ \Rightarrow 10^{x}=\frac{1}{10}$
$ \Rightarrow 10^{x}=10^{-1}$
$ \Rightarrow x=-1 \ldots \ldots .\left[\text { If } a^m=a^n \text {; then } m=n\right]$
$ \therefore \log _{10} 0.1=-1$
View full question & answer→Question 292 Marks
Find the logarithm of : $100$ to the base $10$
AnswerLet $log_{10}100 = x$
$\therefore 10^x = 100$
$\Rightarrow 10^x = 10 \times 10$
$\Rightarrow 10^x = 10^2$
$\Rightarrow x = 2 .....[\text { If } a^m = a^n \text { ; then } m = n ]$
$\therefore log_{10}100 = 2$
View full question & answer→Question 302 Marks
Express the following in exponential form $: \log_{10}1 = 0$
Answer$\log_{10}1 = 0$
$\Rightarrow 10^0 = 1 .....[ \log_ac = b \Rightarrow a^b = c ]$
View full question & answer→Question 312 Marks
Express the following in exponential form :$\log_aA= x$
Answer$\log_aA= x$
$\Rightarrow a^x= A .....[ \log_ac = b \Rightarrow a^b = c ]$
View full question & answer→Question 322 Marks
Express the following in exponential form :$\log_{10}0.01 = - 2$
Answer$\log_{10}0.01 = - 2$
$\Rightarrow 10^{-2} = 0.01 .....[ \log_ac = b \Rightarrow a^b = c ]$
View full question & answer→Question 332 Marks
Express the following in exponential form : $\log_80.125 = -1$
Answer$\log_80.125 = -1$
$\Rightarrow 8^{-1} = 0.125 ....[\log_ac = b \Rightarrow a^b = c ]$
View full question & answer→Question 342 Marks
Express the following in logarithmic form $: (81)^{\frac{3}{4}}=27$
Answer$(81)^{\frac{3}{4}}=27$
$\Rightarrow \log _{81} 27=\frac{3}{4} \quad \ldots .\left[\right.$ By definition of logarithm, $a^b=c \Rightarrow \log _a c= b ]$
View full question & answer→Question 352 Marks
Express the following in logarithmic form :$10^{-3}= 0.001$
Answer$10^{-3}= 0.001$
$\Rightarrow \log_{10}0.001$
$= - 3 ....[ a^b = c \Rightarrow \log_ac = b ]$
View full question & answer→Question 362 Marks
If $\log (x^2- 21) = 2,$ show that $x =± 11.$
Answer$ \log \left(x^2-21\right)=2 $
$\Rightarrow x^2-21=10^2 $
$\Rightarrow x^2-21=100 $
$\Rightarrow x^2=121 $
$\Rightarrow x= \pm 11$
View full question & answer→Question 372 Marks
Express the following in logarithmic form $ :3^{-2}=\frac{1}{9}$
Answer$ 3^{-2}=\frac{1}{9}$
$ \Rightarrow \log _3 \frac{1}{9}$
$=-2 \ldots .\left[a^b=c \Rightarrow \log _a c=b\right]$
View full question & answer→Question 382 Marks
Express the following in logarithmic form $: 5^3= 125$
Answer$5^3= 125$
$\Rightarrow \log_5125$
$= 3 ...[ a^b = c \Rightarrow \log_ac = b ]$
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