Find the logarithmic

Question

To find the $\log $ of $a=be^c $ I am not sure but I suppose this is how it goes .. $\log a = \log b + c \log e $ i.e. $\log a=\log b + c $ But I am a bit confused about the base quantity What comes at base Is it $10 $ or $e$ ??

Answer 1

It's stated in the comments that your equation assumes that the base of the logarithm is $e$. I'd like to expand on them and provide a method for what to do when that isn't true.

First, notice that once you took the logarithm of both sides you had:

$$log(a) = log(be^c)$$

You used the rules of logarithms but skipped too many steps. You should get:

$$log(a) = log(b) + log(e^c) = log(b) + clog(e)$$

This is about as far as you can get if you don't know what the base of the logarithm is. However, you turned it into:

$$log(a) = log(b) + log(e^c) = log(b) + c$$

This implicitly assumes that the base must be $e$, since you let $log(e) = 1$. Since you can't take logarithms of an equation with mixed bases without appropriately changing the base, the answer is that your logarithms are all base $e$.

On the other hand, if your question explicitly stated that the logarithm had to be base 10, and you'd like to further simplify, then you'd have to change the base. In order to do that, you would use the formula:

$$log_b(a) = \frac{log_c(a)}{log_c(b)}$$

In this case, it would mean dividing both sides by $log_{10}(e)$ in order to change the base into base $e$.