Chapter 1 - Functions - 1.3 Inverse, Exponential, and Logarithmic Functions - 1.3 Exercises - Page 38: 84

See proof

We have to prove the statement: $\log_b xy=\log_b x+\log_b y$ a) Let $x=b^p$ $y=b^q$ Solve these two equations for $p$ and $q$: $\log_b x=\log_b b^p$ $\log_b x=p\log_b b$ $\log_b x=p$ $\log_b y=\log_b b^q$ $\log_b y=q\log_b b$ $\log_b x=q$ b) Use the property E1 for exponents to express $xy$ in terms of $b,p,q$: $xy=b^p\cdot b^q$ $xy=b^{p+q}$ c) Compute $\log_b xy$ and simplify: $\log_b xy=\log_b b^{p+q}$ $\log_b xy=(p+q)\log_b b$ $\log_b xy=p+q$ $\log_b xy=\log_b x+\log_b y$