Solve e^(x + 1) = 10 Step by Step

Step by step

The question

Question: solve e to the power (x + 1) = 10.

x is up in the power.

e^{x + 1} = 10

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 + 1}) = ln 10

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 10 ÷ 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 + 1.

x + 1 = ln 10

3 Remove the + 1

We want x by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

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

x ~~+ 1 − 1~~ = ln 10 − 1

+ 1 and − 1 make zero.

Cross them out.

So x = ln 10 − 1.

This is the exact answer.

x = ln 10 − 1

4 Use a calculator

With a calculator, ln 10 is about 2.3026.

Then 2.3026 − 1 is about 1.303.

ln 10 ≈ 2.3026

2.3026 − 1 ≈ 1.303

So x is about 1.303.

x ≈ 1.303

So x = ln 10 − 1, about 1.303 to three decimal places.

5 Check

Let's check with a calculator.

The power gives about 10.0.

That is 10.

It works!

Check: e^{(1.303) + 1} ≈ 10.0

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