Step by step
The question
Question: solve e to the power (2x + 1) = 9.
x is up in the power.
e is a special number, like π.
It is about 2.718.
A log with base e is written ln.
We say "natural log".
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.
Careful!
x is up in the power.
Dividing by e does not bring it down.
Only a log does.
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.
So the left side is just 2x + 1.
3 Remove the + 1
We want 2x 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.
+ 1 and − 1 make zero.
Cross them out.
We keep it exact: 2x = ln 9 − 1.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
So x = (ln 9 − 1)/2.
This is the exact answer.
5 Use a calculator
With a calculator, ln 9 is about 2.1972.
Then (2.1972 − 1) ÷ 2 is about 0.599.
So x is about 0.599.
So x = (ln 9 − 1)/2, about 0.599 to three decimal places.
6 Check
Let's check with a calculator.
The power gives about 9.0.
That is 9.
It works!
Let's look back.
First: take ln of both sides.
Then: ln and e undo each other.
Then: remove the + 1.
Then: remove the 2.
Then: use a calculator.
Last: check.