Step by step
The question
Question: solve 3 times 2 to the power x = 45.
x is up in the power.
Careful!
The power is only on the 2.
Powers come before times.
So we can't join 3 and 2 first.
1 Get the power alone
First the power must be alone: nothing added to it, nothing times it.
The power is times 3.
To remove times 3, we divide by 3.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
3 divided by 3 is 1.
So just the power is left.
45 divided by 3 is 15.
Now we have 2ˣ = 15.
2 Compare with powers of 2
2³ is 8, and 2⁴ is 16.
15 is between them.
So x is between 3 and 4: not a whole number.
3 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.
4 Bring the power down
The power rule for logs: the power comes down in front, as times.
So log of 2ˣ is x times log 2.
5 Remove the log 2
x times log 2.
To remove times log 2, we divide by log 2.
We do it on both sides, so they stay equal.
log 2 divided by log 2 is 1.
So just x is left.
So x = log 15 over log 2.
This is the exact answer.
6 Use a calculator
With a calculator: log 15 is about 1.1761.
Log 2 is about 0.3010.
1.1761 divided by 0.3010 is about 3.907.
So x is about 3.907, to three decimal places.
It is between 3 and 4, as we said.
7 Check
Let's check in the question: put 3.907 in place of x.
With a calculator.
We get about 45.0.
That is 45.
It works!
Let's look back.
First: get the power alone.
Then: compare with powers of 2.
Then: take the log of both sides.
Then: bring the power down.
Then: remove the log 2.
Then: use a calculator.
Last: check.