Solve e^x = 5 Step by Step | Natural logarithms

Step by step

The question

Question: solve e to the power x = 5.

x is up in the power.

e^x = 5

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 5

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 5 ÷ 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 5.

This is the exact answer.

x = ln 5

3 Use a calculator

With a calculator, ln 5 is about 1.6094.

ln 5 ≈ 1.6094

So x is about 1.609.

x ≈ 1.609

So x = ln 5, about 1.609 to three decimal places.

4 Check

Let's check with a calculator.

e to the power 1.609 gives about 5.0.

That is 5.

It works!

Check: e^1.609 ≈ 5.0

Let's look back.

First: take ln of both sides.

Then: ln and e undo each other.

Then: use a calculator.

Last: check.

Answer