Step by step
The question
Question: solve ln(x − 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(x − 1) = 3 means: e to the power 3 is x − 1.
So x − 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 x 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.
So x = e³ + 1.
This is the exact answer.
3 Use a calculator
With a calculator, e to the power 3 is about 20.0855.
Then 20.0855 + 1 is about 21.086.
So x is about 21.086.
So x = e³ + 1, about 21.086 to three decimal places.
4 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: use a calculator.
Last: check.