Step by step
The question
Question: solve 3 times e to the power x = 21.
x is up in the power.
1 Get the power alone
First the power must be alone: nothing added to it, nothing times it.
The power is times 3.
To remove times 3, we divide by 3.
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.
3 divided by 3 is 1.
So just the power is left.
21 divided by 3 is 7.
Now we have eˣ = 7.
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.
Careful!
x is up in the power.
Dividing by e does not bring it down.
Only a log does.
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.
So x = ln 7.
This is the exact answer.
4 Use a calculator
With a calculator, ln 7 is about 1.9459.
So x is about 1.946.
So x = ln 7, about 1.946 to three decimal places.
5 Check
Let's check in the question: put 1.946 in place of x.
With a calculator.
We get about 21.0.
That is 21.
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.