Chapter 0.8, Problem 1E

To determine

To fill: The following table,

Exponent Form	${10}^2=100$	$4^3=64$	$4^4=256$	${0.45}^0=1$	$8^{\frac{1}{2}}=2\sqrt{2}$	$4^{-3}=\frac{1}{64}$
Logarithmic Form

Answer to Problem 1E

Solution:

The complete table is,

Exponent Form	${10}^2=100$	$4^3=64$	$4^4=256$	${0.45}^0=1$	$8^{\frac{1}{2}}=2\sqrt{2}$	$4^{-3}=\frac{1}{64}$
Logarithmic Form	${\log }_{10}100=2$	${\log }_{4}64=3$	${\log }_{4}256=4$	${\log }_{0.45}1=0$	${\log }_{8}2\sqrt{2}=\frac{1}{2}$	${\log }_4\frac{1}{64}=-3$

Explanation of Solution

Given information:

The given table is,

Exponent Form	${10}^2=100$	$4^3=64$	$4^4=256$	${0.45}^0=1$	$8^{\frac{1}{2}}=2\sqrt{2}$	$4^{-3}=\frac{1}{64}$
Logarithmic Form

Consider the exponent, ${10}^2=100$

The exponent form $a^c=b$ means logarithmic form, ${\log }_{a}b=c$.

Then the exponent form ${10}^2=100$ mean logarithmic form ${\log }_{10}100=2$.

Consider the exponent, $4^3=64$

Since the exponent form $a^c=b$ means logarithmic form, ${\log }_{a}b=c$.

Then the exponent form $4^3=64$ mean logarithmic form ${\log }_{4}64=3$.

Consider the exponent, $4^4=256$

Since the exponent form $a^c=b$ means logarithmic form, ${\log }_{a}b=c$.

Then the exponent form $4^4=256$ mean logarithmic form ${\log }_{4}256=4$.

Consider the exponent, ${0.45}^0=1$

Since the exponent form $a^c=b$ means logarithmic form, ${\log }_{a}b=c$.

Then the exponent form ${0.45}^0=1$ mean logarithmic form ${\log }_{0.45}1=0$.

Consider the exponent, $8^{\frac{1}{2}}=2\sqrt{2}$

Since the exponent form $a^c=b$ means logarithmic form, ${\log }_{a}b=c$.

Then the exponent form $8^{\frac{1}{2}}=2\sqrt{2}$ mean logarithmic form ${\log }_{8}2\sqrt{2}=\frac{1}{2}$.

Consider the exponent, $4^{-3}=\frac{1}{64}$

Since the exponent form $a^c=b$ means logarithmic form, ${\log }_{a}b=c$.

Then the exponent form $4^{-3}=\frac{1}{64}$ mean logarithmic form ${\log }_4\frac{1}{64}=-3$.

Hence, the complete table is

Exponent Form	${10}^2=100$	$4^3=64$	$4^4=256$	${0.45}^0=1$	$8^{\frac{1}{2}}=2\sqrt{2}$	$4^{-3}=\frac{1}{64}$
Logarithmic Form	${\log }_{10}100=2$	${\log }_{4}64=3$	${\log }_{4}256=4$	${\log }_{0.45}1=0$	${\log }_{8}2\sqrt{2}=\frac{1}{2}$	${\log }_4\frac{1}{64}=-3$