Step by step
The question
Question: solve 5 to the power (2x − 1) = 125.
x is up in the power.
1 Write 125 as a power of 5
125 is 5 times itself, 3 times.
So 125 is 5³.
Now both sides are 5 to a power.
2 The powers are equal
Both sides are 5 to a power.
Same base, and they are equal.
So the powers must be equal too.
Now we have 2x − 1 = 3.
We want x by itself.
Two things are stuck to x: times 2 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 1
To remove − 1, we add 1.
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.
− 1 and + 1 make zero.
Cross them out.
3 + 1 = 4.
Now we have 2x = 4.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
4 divided by 2 is 2.
So x = 2!
5 Check
Let's check: put 2 in place of x.
We get 125, the same as the right side.
So x = 2 is right!
Let's look back.
First: write 125 as a power of 5.
Then: the powers are equal.
Then: remove the − 1.
Then: remove the 2.
Last: check.