## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Assume the opposite, that both gave the same number $\log_{a}c=\log_{c}a=x$, then use the definition. $\log_{a}c=x$ means that x is the exponent needed to raise a in order to obtain c, $a^{x}=c$ $\log_{c}a=x$ means that x is the exponent needed to raise c in order to obtain a, $c^{x}=a.$ Substitute this a in the equation above: $(c^{x})^{x}=c$ $c^{x^{2}}=c^{1}\quad\Rightarrow\quad x^{2}=1.$ This only works for $x=1$ or $x=-1$, that is if $a=c$ or $a=\displaystyle \frac{1}{c}.$ For any other combination of a and c, it follows that $\log_{a}c\neq\log_{c}a$