SChand CLASS 9 Chapter 7 Logarithms TEST

 TEST

Question 1

The value of $\log_2 16$ is

(a) $\frac{1}{8}$

(b) 4

(c) 8

(d) 16

Sol :

(b) 4

$\log_2 16 = \log_2 2^4 = 4\log_2 2$ = 4×1 (∵ $log_2 a = 1$)

= 4

Question 2

If $a^x = b^y$ then

(a) $\log \frac{a}{b}=\frac{x}{y}$

(b) $\frac{\log a}{ \log b}=\frac{x}{y}$

(c) $\frac{\log a}{\log b}=\frac{y}{x}$

(d) None of these

Sol :

$a^x = b^y$

Taking log both sides,

$log_a x = b_by $

⇒ $x\log_a = y\log_b$

⇒ $\frac{\log a}{\log b}=\frac{y}{x}$

Question 3

If log 3 = 0.477 and $(1000)^x = 3$, then x equals

(a) 0.0159

(b) 0.0477

(c) 0.159

(d) 10

Sol :

log 3 = 0.477

$(1000)^x = 3$

Taking log of both sides

xlog 1000 = log3

⇒ x × 3 = log3 (∵ log 1000 = 3)

⇒ 3x = 0.477 

⇒ $x = \frac{0.477}{3}$

x = 0.159

Question 4

If $\log_{10}2 = 0.3010$, the value of log105 is

(a) 0.3241

(b) 0.6911

(c) 0.6990

(d) 0.7525

Sol :

$\log_{10} 2 = 0.3010$

$\log_{10} 5 = \log_{10} \left(\frac{10}{2}\right) = \log_{10} 10 – \log_{10} 2$

= 1 – 0.3010 = 0.6990 (c)

Question 5

If $\log_{10} 2 = 0.3010$, the value of $\log_{10} 80$ is

(a) 1.6020

(b) 1.9030

(c) 3.9030

(d) None of these

Sol :

(b) 1.9030

$\log_{10} 2 = 0.3010$

$\log_{10} 80 =\log_{10} 10 \times 2^3$

$= \log_{10} 10 + \log_{10} 2^3$

$= \log_{10} 10 + 3\log_{10} 2$

= 1 + 3×0.3010

= 1 + 0.9030 = 1.9030

Question 6

If $\log_{10} 7 = a$, then $\log_{10} \left(\frac{1}{70}\right)$ is equal to

(a) – (1 + a)

(b) $(1 + a)^{-1}$

(c) $\frac{a}{10}$

(d) $\frac{1}{10a}$

Sol :

(a) – (1 + a)

$\log_{10} 7 = a$

$\log_{10} \left(\frac{1}{70}\right) = \log_{10} 1 – \log_{10} 70$

$= 0 – \log_{10} (7 \times 10)$

$= 0 – \log_{10} 7 – \log_{10} 10$

= 0 – 0 – 1 = – (1 + a)

Question 7

If log 27 = 1.431, then the value of log 9 is

(a) 0.934

(b) 0.945

(c) 0.954

(d) 0.958

Sol :

(c) 0.954

log 27 = 1.431 

⇒ log 3³ = 1.431

⇒ 31og 3 = 1.431

⇒ log 3 $= \frac{1.431}{3}$ = 0.477

log 9 = log 3² = 2 log 2

= 2×0.477 = 0.954

Question 8

If $\log_{10}  5 + \log_{10}(5x + 1) = \log_{10}(x + 5) + 1$, then x is equal to

(a) 1

(b) 3

(c) 5

(d) 10

Sol :

(b) 3

$\log_{10} 5 + \log_{10}(5x + 1) = log_{10}(x + 5) + 1$

$\log_{10}5 (5x + 1) = \log_{10} (x + 5) + \log_{10} 10$

$\log_{10}(5x + 1) = \log_{10} 10 (x + 5)$

Comparing, we get

⇒ 5(5x + 1) = 10(x + 5)

⇒ 25x + 5 = 10x + 50

⇒ 25x – 10x = 50 – 5

⇒ 15x = 45 

⇒ $x = \frac{45}{15} = 3$

⇒ x = 3

Question 9

If $\log_x 4 = 0.4$, then the value of x is

(a) 1

(b) 4

(c) 16

(d) 32

Sol :

$\log_x 4 = 0.4$

⇒ $4 = x^{0.4}$

⇒ $x = 4^{1}{0.4}=(2^2)^{\frac{1}{0.4}}$

= $2^{2×\frac{1}{0.4}}=2^{\frac{1}{0.2}}$

= $2^{\frac{1}{5}}=2^{1×\frac{5}{1}}=2^5$

= 2×2×2×2×2 = 32

∴ x = 32

Question 10

The solution of $\log_π [\log_2 (\log_7 x)] = 0 $ is

(a) 2

(b) π²

(c) 72

(d) None of these

Sol :

$\log_π [\log_2 (\log_7 x)] = 0$

⇒ $\log_2 (log_7 x) = π° = 1$

⇒ $\log_7 x = 2^1 = 2$

⇒ $x = 7^2$

∴ $x = 7^2$