Step by step
The question
Question: solve e to the power (3x) = 20.
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 3x.
3 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 = ln 20/3.
This is the exact answer.
4 Use a calculator
With a calculator, ln 20 is about 2.9957.
Then 2.9957 ÷ 3 is about 0.999.
So x is about 0.999.
So x = ln 20/3, about 0.999 to three decimal places.
5 Check
Let's check with a calculator.
The power gives about 20.0.
That is 20.
It works!
Let's look back.
First: take ln of both sides.
Then: ln and e undo each other.
Then: remove the 3.
Then: use a calculator.
Last: check.