Step by step
The question
Question: write 2 log x + log 5 as one log.
2 log x + log 5
The three log rules, used backwards.
A number in front goes up as a power.
Plus becomes times.
Minus becomes divide.
n × log a = log(an)
log a + log b = log(a × b)
log a − log b = log(ab)
1 Numbers in front go up
Careful!
The 2 in front is a power, not times.
2 log x is log of x to the power 2.
Careful: not 2log x = log(2x) ✗
Each number in front goes up as a power: log x².
= log x2 + log 5
2 Plus becomes times
Plus between logs becomes times inside: 5 and x² multiplied, the number first.
log a + log b = log(a × b)
So it is one log: log(5x²).
= log(5x2)
3 Check
Let's check with numbers: x = 3.
With a calculator, the question is about 1.653.
x = 3: question ≈ 1.653
Our answer: the inside is 45, and its log is about 1.653.
The same.
It works!
answer: log 45 ≈ 1.653
Let's look back.
First: numbers in front go up.
Then: plus becomes times.
Last: check.
Answer
= log(5x2)