Step by step
The question
Question: solve 3ˣ = 20.
x is up in the power.
1 Compare with powers of 3
3² is 9, and 3³ is 27.
20 is between them.
So x is between 2 and 3: not a whole number.
2 Take the log of both sides
Logs can bring a power down.
So we take the log of both sides.
We do it on both sides, so they stay equal.
3 Bring the power down
The power rule for logs: the power comes down in front, as times.
So log of 3ˣ is x times log 3.
4 Remove the log 3
x times log 3.
To remove times log 3, we divide by log 3.
We do it on both sides, so they stay equal.
log 3 divided by log 3 is 1.
So just x is left.
So x = log 20 over log 3.
This is the exact answer.
5 Use a calculator
With a calculator: log 20 is about 1.3010.
Log 3 is about 0.4771.
1.3010 divided by 0.4771 is about 2.727.
So x is about 2.727, to three decimal places.
It is between 2 and 3, as we said.
6 Check
Let's check with a calculator: 3 to the power 2.727 is about 20.0.
That is 20.
It works!
Let's look back.
First: compare with powers of 3.
Then: take the log of both sides.
Then: bring the power down.
Then: remove the log 3.
Then: use a calculator.
Last: check.