Step by step
The question
Question: solve e to the power (x − 2) = 7.
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 x − 2.
3 Remove the − 2
We want x by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 make zero.
Cross them out.
So x = ln 7 + 2.
This is the exact answer.
4 Use a calculator
With a calculator, ln 7 is about 1.9459.
Then 1.9459 + 2 is about 3.946.
So x is about 3.946.
So x = ln 7 + 2, about 3.946 to three decimal places.
5 Check
Let's check with a calculator.
The power gives about 7.0.
That is 7.
It works!
Let's look back.
First: take ln of both sides.
Then: ln and e undo each other.
Then: remove the − 2.
Then: use a calculator.
Last: check.