When do we use common logarithms and when do we use natural logarithms

Question

Currently, in my math class, we are learning about logarithms. I understand that the common logarithm has a base of 10 and the natural has a base of e. But, when do we use them?

For example the equation $7^{x-2} = 30$ in the lesson, you solve by rewriting the equation in logarithmic form $\log_7 30 = x-2$. The,n apply the change of base formula, and use a calculator to evaluate.

$$\frac{\ln30}{\ln7}$$

now this is where I get confused. Why do use natural logarithms here? Why don't we use common logarithms? Am I missing something simple?

Any help is greatly appreciated.

Answer 1

The point of making a change of base is that your calculator probably doesn't have a button to evaluate logarithms with an arbitrary base, but it does have a button to evaluate natural logarithms. So the only thing special about $e$ here is that your calculator knows how to compute logarithms in base $e$. If your calculator happens to also have a button for logarithms in base $10$, it would be perfectly fine to use them instead.

Answer 2

The answer for your question about solving $7^{x-2}=30$ is that you dont have to use the natural logarithm. You can use the base 10 logarithm and you'll get the same answer. That's how you get the change of basis formula.

We tend to use the natural logarithm because $e$ comes up quite often in certain formulas, equations, etc, and the natural logarithm cancels it out. $e$ tends to come up more than 10, so it just gets used by default.