Solve e^x = 30 Step by Step | Natural logarithms

Step by step

The question

Question: solve e to the power x = 30.

x is up in the power.

e^x = 30

e is a special number, like π.

It is about 2.718.

A log with base e is written ln.

We say "natural log".

e ≈ 2.718

ln = log_e

1 Take ln of both sides

To bring the power down, we use a log.

With e, we use ln.

We do it on both sides, so they stay equal.

ln(e^x) = ln 30

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 30 ÷ e ✗

2 ln and e undo each other

ln asks: e to what power?

For e to the power k, the answer is k.

So ln and e undo each other.

ln(e^k) = k

So x = ln 30.

This is the exact answer.

x = ln 30

3 Use a calculator

With a calculator, ln 30 is about 3.4012.

ln 30 ≈ 3.4012

So x is about 3.401.

x ≈ 3.401

So x = ln 30, about 3.401 to three decimal places.

4 Check

Let's check with a calculator.

e to the power 3.401 gives about 30.0.

That is 30.

It works!

Check: e^3.401 ≈ 30.0

Let's look back.

First: take ln of both sides.

Then: ln and e undo each other.

Then: use a calculator.

Last: check.

Answer