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.
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