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Question 512 Marks

Find $x$ in the following when: $\log _{10} 0.0001 = x$

Answer

$\log _{10} 0.0001 = x$
$\Rightarrow 0.0001 = 10^x$
$\Rightarrow 10^x = 10^{-4}$
$\therefore x = -4.$

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Question 522 Marks

Find $x$ in the following when$: \log _3 243 = x$

Answer

$\log _3 243 = x$
$\Rightarrow 243 = 3^x$
$\Rightarrow 3^5 = 3^x$
$\therefore x = 5.$

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Question 532 Marks

Find $x$ in the following when $: \log_2 \ 0.5 = x$

Answer

$\log _2 0.5=x$
$\Rightarrow 2^x=0.5=\frac{1}{2}$
$\Rightarrow 2^x=2^{-1}$
$\therefore x=-1 .$

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Question 542 Marks

Find $x$ in the following when$: \log _4 x = -4$

Answer

$\log _4 x=-4$
$\Rightarrow x=4^{-4}$
$ \Rightarrow x=\frac{1}{256} .$

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Question 552 Marks

Find $x$ in the following when: $\log _7 x = 3$

Answer

$\log _7 x = 3$
$\Rightarrow x = 7^3$
$\Rightarrow x = 343.$

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Question 562 Marks

Find $x$ in the following when$: \log _8 x=\frac{2}{3}$

Answer

$\log _8 x=\frac{2}{3}$
$\Rightarrow x=8^{\frac{2}{3}}$
$ \Rightarrow x^3=8^2$
$=64$
$=4^3 $
$ \therefore x=4 .$

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Question 572 Marks

Find $x$ in the following when$: \log _x 243 = 5$

Answer

$\log _x 243 = 5$
$\Rightarrow x^5 = 243$
$\Rightarrow x^5 = 3^5$
$\therefore x = 3.$

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Question 582 Marks

Find $x$ in the following when$: \log_x 125 = 3$

Answer

$\log_x 125 = 3$
$\Rightarrow x^3 = 125$
$\Rightarrow x^3 = 5^3$
$\therefore x = 5.$

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Question 592 Marks

Find $x$ in the following when$: \log _x 49=2$

Answer

$\log _x 49 = 2$
$\Rightarrow x^2 = 49$
$\Rightarrow x^2 = 7^2$
$\therefore x = 7.$

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Question 602 Marks

Express the following in the exponential form$:\log _{\sqrt{6}}(6 \sqrt{6})=3$

Answer

$\log _{\sqrt{6}}(6 \sqrt{6})=3$
$ \Rightarrow 6 \sqrt{6}$
$=(\sqrt{6})^3 .$

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Question 612 Marks

Express the following in the logarithmic form$:9^{-4}=\frac{1}{6561}$

Answer

$9^{-4}=\frac{1}{6561}$
$\Rightarrow \log _9 \frac{1}{6561}$
$=-4$

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Question 622 Marks

Express the following in the logarithmic form$:\left(\frac{1}{3}\right)^4=\frac{1}{81}$

Answer

$\left(\frac{1}{3}\right)^4=\frac{1}{81}$
$ \Rightarrow \log _{\frac{1}{3}} \frac{1}{81}$
$=4$

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