How to compute the exponent?

Question

So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$?

Thanks in advance!

Answer 1

Apply Logarithm on both sides:

$$ \log a^n = \log b$$ $$n \log a = \log b$$ $$n = \frac{\log b}{\log a}$$

If you are a starter in Logarithms, you can refer here.

Answer 2

$$ \begin{align}a^n &= b\\ \Rightarrow \log_{a}{a^n} &= \log_{a}{b}\quad(\because \log_{a}a^n = n)\\ \Rightarrow n &= \log_{a}{b} \end{align} $$

Answer 3

$$ a^n = b $$ $$ log_{a}b = n $$

Because the easily accessible log button on your calculator is probably base 10 and not base a, you have to punch it in this way:

$$\frac {\log b} {\log a}$$

which will result in your answer, $n$.

If you have a TI-89 Titanium, Diamond 7 is the way to quickly access the log function (it took me a long time to find this).

Answer 4

Exponent problems like finding $n$ if we know the value of both $a$ and $b$ in the equation $a^n = b$ can be solved using logarithms:

$$ \begin{align} a^n=b&\Rightarrow \log \left(a^n\right)=\log (b) \\ &\Rightarrow n\log(a)=\log(b) \\ &\Rightarrow n=\dfrac{\log(b)}{\log(a)}=\log_a (b) \end{align}$$