Laws of logarithms, step by step
Adding two logarithms multiplies their numbers; subtracting them divides.
The question
2 log₂ 3
2 log₂ 3
k log₂ m = log₂ (mᵏ)
3² = 9
2 log₂ 3=log₂ 9
- Both logarithms are to the same base, 2. That is what makes a law available at all — two logarithms to different bases cannot be combined, and checking the bases match is the first thing to do rather than the last.
- A logarithm MULTIPLIED by a number is the logarithm of the argument RAISED to that number. The multiplier comes down off the front and goes up as a power — which is why the answer grows so much faster than the question looks like it should.
- So the two arguments become one: 3² = 9. The answer is log₂ 9.
How it works.
- 01
Both logarithms are to the same base, 2. That is what makes a law available at all — two logarithms to different bases cannot be combined, and checking the bases match is the first thing to do rather than the last.
- 02
A logarithm MULTIPLIED by a number is the logarithm of the argument RAISED to that number. The multiplier comes down off the front and goes up as a power — which is why the answer grows so much faster than the question looks like it should.
- 03
So the two arguments become one: 3² = 9. The answer is log₂ 9.