Chapter 1 - Functions - 1.3 Inverse, Exponential, and Logarithmic Functions - 1.3 Exercises - Page 38: 86

See proof

We have to prove the statement: $\log_b x^z=z\log_b x$ a) Let $x=b^p$ Solve for $p$: $\log_b x=\log_b b^p$ $\log_b x=p\log_b b$ $\log_b x=p$ b) Use the property E3 for exponents to express $x^z$ in terms of $b,p$: $x^z=(b^p)^z$ $x^z=b^{pz}$ c) Compute $\log_b x^z$ and simplify: $\log_b x^z=\log_b b^{pz}$ $\log_b x^z=pz\log_b b$ $\log_b x^z=pz$ $\log_b x^z=z\log_b x$