# How would you solve different logs?

## What is log ${\log}_{5} \wedge {5}^{7}$

Apr 25, 2018

7

#### Explanation:

1. For this question:
The easiest way to think about logs for most people is to think of them in terms of their exponential inverses.
For example, ${\log}_{10} 100 = 2$ because the base $\left(10\right)$ raised to the answer $\left(2\right)$ equals the thing you're taking the log of $\left(100\right)$. As an equation, ${\log}_{10} 100 = 2$ because ${10}^{2} = 100$.
Similarly, ${\log}_{2} 8 = 3$ because ${2}^{3} = 8$.
Looking back at your question, let's give it an answer $x$.
Then ${\log}_{5} {5}^{7} = x$. Putting that in exponential form, ${5}^{7} = {5}^{x}$, makes it clear that $x = 7$.

2. For other, non-standard base logs, use the change of base formula. Divide the log of the big number (here it's ${5}^{7}$) by the log of the little number ($5$). So $\frac{\log {5}^{7}}{\log 5} = 7$. You can even use natural logs for this function! $\frac{\ln {5}^{7}}{\ln 5} = 7$ too.

3. I find it handy to keep in mind the two examples I gave earlier when I'm thinking about logarithms: ${\log}_{10} 100 = 2$ and ${\log}_{2} 8 = 3$. Writing formulas you know at the top of your paper will help you think through new problems.

Hope this helps, and Happy mathing!