Solve e^(-x) = 4 Step by Step | Natural logarithms

Step by step

The question

Question: solve e to the power (−x) = 4.

x is up in the power.

e^−x = 4

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 4

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 4 ÷ 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 the left side is just −x.

−x = ln 4

3 Remove the −1

−x means −1 times x.

We want just x.

To remove times −1, we divide by −1.

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

−x−1 = ln 4−1

−1 divided by −1 is 1.

So just x is left.

ln 4 divided by −1 is −ln 4: the sign changes.

So x = −ln 4.

This is the exact answer.

x = −ln 4

4 Use a calculator

With a calculator, ln 4 is about 1.3863.

Then −1.3863 is about −1.386.

ln 4 ≈ 1.3863

−1.3863 ≈ −1.386

So x is about −1.386.

x ≈ −1.386

So x = −ln 4, about −1.386 to three decimal places.

5 Check

Let's check with a calculator.

The power gives about 4.0.

That is 4.

It works!

Check: e^{−(−1.386)} ≈ 4.0

Let's look back.

First: take ln of both sides.

Then: ln and e undo each other.

Then: remove the −1.

Then: use a calculator.

Last: check.

Answer