## University Calculus: Early Transcendentals (3rd Edition)

(a) $2^{\log_4x}=\sqrt x$ (b) $9^{\log_3x}=x^2$ (c) $\log_2(e^{(\ln2)(\sin x)})=\sin x$
(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 $$\log_2(e^{(\ln2)(\sin x)})=\sin x$$