Removing logs from equation

Question

I have a simple question that I need clarification on:

If

$$\log(a) = \log(b) + c$$

is it true that

$$a = b + \exp(c)$$

Is this correct or am I missing something really basic that I cant remember from maths class?

Answer 1

You’re missing something, namely, one of the laws of exponents: if $\log a=\log b+c$, where these are natural logs, then

$$a=e^{\log a}=e^{\log b+c}=e^{\log b}\cdot e^c=be^c\;.$$

(If they are common logs, with base $10$ rather than $e$, replace every $e$ by $10$.)

Answer 2

HINT:

Assuming the base to be $b$ we can write $c=\log_bb^c$

Use $\log A+\log B=\log AB$ and for real $C,D$ $$\log_bC=\log_bD\iff C=D$$

Answer 3

You're missing something. Here's how it should look:

$$ \begin{aligned} log(a) &= log(b)+c \\ e^{log(a)} &= e^{log(b)+c} \\ a &= e^{log(b)}e^{c} \\ a &= b\cdot e^c \end{aligned} $$