Solve e^x + 4 = 10 Step by Step | Exponential equations

Step by step

The question

Question: solve e to the power x, + 4 = 10.

x is up in the power.

e^x + 4 = 10

1 Get the power alone

First the power must be alone: nothing added to it, nothing times it.

first: the power alone

We want the power by itself.

The + 4 is in the way.

To remove + 4, we take away 4.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

e^x ~~+ 4 − 4~~ = 10 − 4

+ 4 and − 4 make zero.

Cross them out.

10 − 4 is 6.

Now we have eˣ = 6.

e^x = 6

2 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 6

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 6 ÷ e ✗

3 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 6.

This is the exact answer.

x = ln 6

4 Use a calculator

With a calculator, ln 6 is about 1.7918.

ln 6 ≈ 1.7918

So x is about 1.792.

x ≈ 1.792

So x = ln 6, about 1.792 to three decimal places.

5 Check

Let's check in the question: put 1.792 in place of x.

With a calculator.

Check: e^1.792 + 4 ≈ 6.0 + 4 ≈ 10.0

We get about 10.0.

That is 10.

It works!

Let's look back.

First: get the power alone.

Then: take ln of both sides.

Then: ln and e undo each other.

Then: use a calculator.

Last: check.

Answer