Question #e10e6

Jul 10, 2016

$m = - 100$

Explanation:

The only way to get the answer

$m = - 100$

is to have

$\log \left(- m\right) + 2 = 4$

as your starting equation. As you know, the common log, which is denoted $\log$, is actually the log base $10$. This means that your starting equation can be rewritten as

${\log}_{10} \left(- m\right) + 2 = 4$

The first thing to do here is isolate the log on one side of the equation by adding $- 2$ to both sides

${\log}_{10} \left(- m\right) + \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} - \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} = 4 - 2$

${\log}_{10} \left(- m\right) = 2$

The log function is actually undefined for negative numbers when working with real numbers. This tells you that $- m \ge 0$, which implies that $m \le 0$.

Now, the log function is the inverse operation to exponentiation. This means that you're looking for a number that is equal to the base, which in your case is $\textcolor{red}{10}$, raised to the power of the result, which is $\textcolor{b l u e}{2}$.

${\log}_{\textcolor{red}{10}} \left(\textcolor{b l u e}{- m}\right) = \textcolor{\mathrm{da} r k g r e e n}{2}$

Can thus be rewritten as

${\textcolor{red}{10}}^{\textcolor{\mathrm{da} r k g r e e n}{2}} = \textcolor{b l u e}{- m}$

Since ${10}^{2} = 100$, you will have

$100 = - m \implies m = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{- 100} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

As predicted, $m \le 0$.