Answer
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 $\log_{a}c\neq\log_{c}a$
Work Step by Step
- no extra steps -
