Answer
(a) $2^{\log_4x}=\sqrt x$
(b) $9^{\log_3x}=x^2$
(c) $\log_2(e^{(\ln2)(\sin x)})=\sin x$
Work Step by Step
(a) $$2^{\log_4x}$$
Here $2$ and $4$ do not match each other to use the inverse properties.
But $2=\sqrt4=4^{1/2}$
That means $$2^{\log_4x}=4^{\frac{1}{2}\log_4x}$$
- Apply Power Rule, we have: $\frac{1}{2}\log_4x=\log_4x^{\frac{1}{2}}$
Thus, $$2^{\log_4x}=4^{\log_4x^{\frac{1}{2}}}=x^{\frac{1}{2}}=\sqrt x$$
(b) $$9^{\log_3x}$$
Here again $9$ and $3$ do not match each other to use the inverse properties.
But $9=3^2$
That means $$9^{\log_3x}=3^{2\log_3x}$$
- Apply Power Rule, we have: $2\log_3x=\log_3x^2$
Thus, $$9^{\log_3x}=3^{\log_3x^2}=x^2$$
(c) $$\log_2(e^{(\ln2)(\sin x)})$$
- First, for $e^{(\ln2)(\sin x)}$, we can apply the Power Rule: $e^{(\ln2)(\sin x)}=e^{\ln(2^{\sin x})}$
- You can see here that we can apply inverse properties with base $e$, meaning
$$e^{(\ln2)(\sin x)}=2^{\sin x}$$
Therefore, $$\log_2(e^{(\ln2)(\sin x)})=\log_22^{\sin x}$$
- We again apply inverse properties with base $a$, meaning
$$\log_2(e^{(\ln2)(\sin x)})=\sin x$$
