What does $ \log_a (b) $ equal to?

Question

Does $$ \log_a(b) = \frac{\log_c (b)}{\log_c (a)}$$ or $$ \log_a(b) = \frac{\ln (b)}{\ln (a)}$$ ??

Is there any difference between the two?

Answer 1

Suppose $$\log_{a} (b) = x.$$

Then, by definition, we have $$b = a^x,$$ where $a>0$.

Now let $c>0$.

Taking the logarithm to the base $c$ of both sides of the equation $b = a^x$, we get $$\log_{c} (b) = \log_{c} (a^x).$$ Or $$\log_{c} (b) = x \log_{c} (a)$$ using the property of the logarithm.

So, if $\log_{c} (a) \ne 0$, then upon dividing both sides of the last equation by $\log_{c} (a) $, we get $$ x = \frac{\log_{c} (b)}{\log_{c} (a)}.$$ Or $$ \log_{a} (b) = \frac{\log_{c} (b)}{\log_{c} (a)}.$$

Now taking $c$ to be equal to $e$ in the last relation, we get $$\log_{a} (b) = \frac{\ln (b)}{\ln (a)}.$$

Answer 2

$\ln(a)$ is just a shorter way to write $\log_e(a)$ so the second formula is an instance of the first with $c=e$, both identities are right