TRUE / FALSE

Question 11 Mark

State, true or false $:\ \log x \times \log y = \log x + \log y$

Answer

We know that
$\log x + \log y = \log xy$
$\therefore \log x + \log y \neq \log x \times \log y$
Thus the statement $\log x + \log y = \log x \times \log y$ is false.

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Question 21 Mark

State, true or false $:\ $ If $\frac{\log 25}{\log 5}=\log x_t$ then $\mathrm{x}=2$

Answer

Given that
$\frac{\log 25}{\log 5}=\log \mathrm{x}$
$ \Rightarrow \frac{\log 5 \times 5}{\log 5}=\log \mathrm{x}$
$ \Rightarrow \frac{\log 5^2}{\log 5}=\log x$
$ \Rightarrow \frac{2 \log 5}{\log 5}=\log x \ldots\left[\log _a m^n=n \log _a m\right]$
$ \Rightarrow 2=\log { }_{10} \mathrm{x}$
$ \Rightarrow 10^2=\mathrm{x}$
$ \Rightarrow \mathrm{x}=100$
Thus, the statement, $x=2$ is false.

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Question 31 Mark

State, true or false $:\ \frac{\log x}{\log y}=\log x-\log y$

Answer

We know that
$\log \left(\frac{m}{n}\right)=\log m-\log n$
$ \therefore \frac{\log x}{\log y} \neq \log x-\log y$
Thus the statement, $\frac{\log x}{\log y}=\log \mathrm{x}-\log \mathrm{y}$ is false.

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Question 41 Mark

State, true or false $:\ \log 1 \times \log 1000 = 0$

Answer

We have,
$\log 1 = 0$ and $\log 1000 = 3$
$\therefore \log 1 \times \log 1000 = 0 \times 3 = 0$
Thus the statement, $\log 1 \times \log 1000 = 0$ is true.

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Question 51 Mark

State, true or false : $\log _2 8=3$ and $\log _8=2=\frac{1}{3}$

Answer

Consider the equation
$
\log _2 8=3
$
$\Rightarrow 2^3=8$ $\ldots(1)$
Now consider the equation
$
\begin{aligned}
& \log _8 2=\frac{1}{3} \\
& \Rightarrow 8^{\frac{1}{3}}=2
\end{aligned}
$
$\Rightarrow\left(2^3\right)^{\frac{1}{3}}=2$ $\ldots(2)$
Both the equations (1) and (2) are correct.
Thus the given statements, $\log _2 8=3$ and $\log _8 2=\frac{1}{3}$ are true.

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Question 61 Mark

State, true or false $:\ $If $x^y= z$, then $y =\log_zx .$

Answer

Consider the equation
$x^y = z$
$\Rightarrow \log_xz = y$
Thus the statement, $\log_zx = y$ is false.

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Question 71 Mark

State, true or false$:\ $If $\log_{10}x = a$, then $10^x= a.$

Answer

Consider the equation
$\log_{10}x = a$
$\Rightarrow 10^a = x$
Thus the statement, $10^x = a$ is false.

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