Answer
(a) $2^{\log_23}=3$
(b) $10^{\log_{10}\frac{1}{2}}=\frac{1}{2}$
(c) $\pi^{\log_{\pi}7}=7$
(d) $\log_{11}121=2$
(e) $\log_{121}11=\frac{1}{2}$
(f) $\log_3\frac{1}{9}=-2$
Work Step by Step
*For (a) to (c): Use the first inverse property with base $a$:
$$a^{\log_a x}=x$$
Therefore:
(a) $$2^{\log_23}=3$$
(b) $$10^{\log_{10}\frac{1}{2}}=\frac{1}{2}$$
(c) $$\pi^{\log_{\pi}7}=7$$
*For (d) to (f): Use the second inverse property with base $a$: $$\log_aa^x=x$$
(d) $$\log_{11}121=\log_{11}11^2$$
- Now apply the inverse property: $$\log_{11}121=2$$
(e) $$\log_{121}11=\log_{121}\sqrt{121}=\log_{121}121^{\frac{1}{2}}$$
- Now apply the inverse property: $$\log_{121}11=\frac{1}{2}$$
(f) $$\log_3\frac{1}{9}=\log_3\frac{1}{3^2}=\log_33^{-2}$$
- Now apply the inverse property: $$\log_3\frac{1}{9}=-2$$
