Answer
(a) $5^{\log_57}=7$
(b) $8^{\log_8\sqrt2}=\sqrt2$
(c) $1.3^{\log_{1.3}75}=75$
(d) $\log_416=2$
(e) $\log_3\sqrt3=\frac{1}{2}$
(f) $\log_4\frac{1}{4}=-1$
Work Step by Step
*For (a) to (c): Use the first inverse property with base $a$:
$$a^{\log_a x}=x$$
Therefore:
(a) $$5^{\log_57}=7$$
(b) $$8^{\log_8\sqrt2}=\sqrt2$$
(c) $$1.3^{\log_{1.3}75}=75$$
*For (d) to (f): Use the second inverse property with base $a$: $$\log_aa^x=x$$
(d) $$\log_416=\log_44^2$$
- Now apply the inverse property: $$\log_416=2$$
(e) $$\log_3\sqrt3=\log_33^{1/2}$$
- Now apply the inverse property: $$\log_3\sqrt3=\frac{1}{2}$$
(f) $$\log_4\frac{1}{4}=\log_44^{-1}$$
- Now apply the inverse property: $$\log_4\frac{1}{4}=-1$$
