Step by step
The question
Question: solve ln(2x − 1) = 3.
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 Write it as a power
A log is a power.
ln(2x − 1) = 3 means: e to the power 3 is 2x − 1.
So 2x − 1 is e³.
Careful!
ln is log base e, not base 10.
So it is e to the power 3, not 10 to the power 3.
2 Remove the − 1
We want 2x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
We keep it exact: 2x = e³ + 1.
3 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 = (e³ + 1)/2.
This is the exact answer.
4 Use a calculator
With a calculator, e to the power 3 is about 20.0855.
Then (20.0855 + 1) ÷ 2 is about 10.543.
So x is about 10.543.
So x = (e³ + 1)/2, about 10.543 to three decimal places.
5 Check
Let's check with a calculator.
The inside is about 20.086.
ln of it is about 3.00.
That is 3.
It works!
Let's look back.
First: write it as a power.
Then: remove the − 1.
Then: remove the 2.
Then: use a calculator.
Last: check.