# Chapter 1 - Section 1.6 - Inverse Functions and Logarithms - Exercises - Page 50: 61

(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: $$\log_4\frac{1}{4}=-1$$

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