Step by step
The question
Question: solve ln(3x) = 2.
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(3x) = 2 means: e to the power 2 is 3x.
So 3x is e².
Careful!
ln is log base e, not base 10.
So it is e to the power 2, not 10 to the power 2.
2 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
So x = e² over 3.
This is the exact answer.
3 Use a calculator
With a calculator, e to the power 2 is about 7.3891.
Then 7.3891 ÷ 3 is about 2.463.
So x is about 2.463.
So x = e² over 3, about 2.463 to three decimal places.
4 Check
Let's check with a calculator.
The inside is about 7.389.
ln of it is about 2.00.
That is 2.
It works!
Let's look back.
First: write it as a power.
Then: remove the 3.
Then: use a calculator.
Last: check.