SChand CLASS 9 Chapter 7 Logarithms Exercise 7(A)

 Exercise 7(A)

Question 1

Give an equivalent exponential form for each statement:

(i) $\log _{2} 8=3$

Sol: $2^{3}=8$

(ii) $\log _{3} 81=4$

Sol : $3^{4}=81$

(iii) $\log _{2} 1 / 2=-1$

Sol :  $1 / 2=2^{-1}$

(iv) $\log _{5} 1 / 25=-2$

Sol :  $\frac{1}{25}=5-2$

(v) $1 / 2=\log _{4} 2$

Sol :  $4^{1 / 2}=2$

(vi) $1 / 3=\log _{27} 3$

Sol :   $27^{1 / 3}=3$

Question 2

Give an equivalent logarithmic form for each statement

(i) $16=2^{4}$

Sol:  Or, $\log _{2} 16=4$

(ii) $25=5^{2}$

Sol:  Or, $\log _{5} 25=2$

(iii) $81=3^{4}$

Sol: Or, $\log _{3} 81=4$

(iv) $6^{0}=1$

Sol:$\log _{6} 1=0$

(v) $8^{1 / 3}$

Sol: $\log _{8} 2=1 / 3$

(vi) $1 / 9=3^{-2}$

Sol: $\log _{3} 1 / 9=-2$

(vii) $1 / 32=2^{-5}$

Sol: $\log_{2}{1 / 32}$=-5

(viii) $10^{1.4969}=31.4$

Sol: $\log _{10} 31 \cdot 4=1.4969$

Question 3

Find the value of each logarithms given below:

(i) $\log _{10} 1000$

$=\log _{10} 10^{3}=3$

(ii) 

$\begin{aligned} & \log _{2} 8 \\=& \log _{2} 2^{3} \\=& 3 \end{aligned}$

(iii) $\log _{3} 81$

$=\log _{3} 3^{4}=4$

(iv) $\log _{10} 0.1$

$=\log _{10} 1 / 10$

=$\log _{10} 10^{-1}$

=-1

(v)

$\begin{aligned} & \log _{10} 0.01 \\=& \log _{10} 1 / 100 \\=& \log _{10} 10-2=-2 \end{aligned}$

(vi) 

$\begin{aligned} & \log _{10} 0.0001 . \\=& \log _{10} 1 / 10000 . \\=& \log _{10} 10^{-4}=-4 \end{aligned}$

(vii)$\log _{2} 1 / 4$

$=\log _{2} 2^{-2}=-2$

(viii)

$\begin{aligned} & \log _{3} 1 / 27 \\=& \log _{3} 3^{-3}=-3 \end{aligned}$

(ix)

$\begin{aligned} & \log _{3} 1 \\=& 0 \end{aligned}$

(x)  $\log _{1 / 2} 1 / 4$

$=\log _{1 / 2} 1 / 2^{2}$

$=2$

(xi) $\log _{27} 9$

$=\frac{\log 9}{\log 27}$

$=\frac{\log 3^{2}}{\log 3^{3}}$

$=\frac{2}{3}$ $\frac{\log 3}{\log 3}$

$=\frac{2}{3}$

(v) $\log _{10} 100=x$

Or, $10^{x}=10^{2}$

$X=2$

(vi) $\log _{2} 0.5=x$ 

$2^{x}=0.5=5 / 10=1 / 2=2^{-1}$

(vii) $\log _{10} x=a$

$10^{\mathrm{a}}=x$

( True )

Question 4

Find the value of x

(i) $log_x 216 = 3$

(ii) $log_4 x = – 4$

(iii) $log_3 x = 0$

(iv) $log_8 x = \frac{2}{3}$

(v) $log_{10} 100 = x$

(vi) $log_2 0.5 = x$

Sol :

(i) $log_x 216 = 3$

⇒ x³ = 216 = (6)³

Comparing, we get

x = 6

(ii) $log_4 x = – 4$

⇒ $(4)^{-4} = x$

⇒ $\frac{1}{4^4} = x$

⇒ $x = \frac{1}{256}$

∴ $x = \frac{1}{256}$

(iii) $log_3 x = 0$

⇒ 3° = x

⇒ x = 1

∴ x = 1

(iv) $log_8 x = \frac{2}{3}$

⇒ $(8)^{2}{3}$

⇒ $x = (2^3)\frac{2}{3}=2^3×\frac{2}{3}$ = 2²

⇒ x = 2×2 = 4

∴ x = 4

(v) $log_{10} 100 = x$

⇒ $10^{x} = 100$

⇒ $10^x = (10)^2$

Comparing, we get

x = 2

(vi) $log_2 0.5 = x$

⇒ $2^x = 0.5 = \frac{1}{2}$

⇒ $2^x = (2)^{-1}$

Comparing both sides

x = – 1

Question 5

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

Sol :

∵ $log_{10} x = a$

⇒ $10^a = x$

Which is true.