Question

How do you solve `e^x>30`?

Answer 1

`x > ln(30)`

Explanation:

It may be helpful to look at a graph of `e^x`. Notice that because it is an exponential function with a base greater than 1, it is monotonically increasing, so:

`e^a > e^b => a > b`

To clarify, I'll use a concrete example.

`e^2 > e^0 => 2 > 0`

Getting back to our original problem, we want to find when `e^x` is greater than `30`. Rewrite `30` as `e^ln(30)`

`e^x > e^ln(30)`

But if `e^x > e^ln(30)`:

`x > ln(30)`

Technically, you could also take the natural logarithm on both sides since that is also monotonically increasing.