Adding two logarithms multiplies their numbers; subtracting them divides. These free printable worksheets drill the three laws in the direction that gets used most — writing a sum, a difference or a multiple as one logarithm.
The three laws are short enough to memorize and easy enough to mix up, which is exactly why a page of them is worth more than a page of any one. Add and you multiply. Subtract and you divide — first over second, and the order matters. Multiply a logarithm by a number and that number goes up as a power. The mistake is almost never the arithmetic; it is reaching for the wrong law.
Why this sheet works
Nothing here can be finished by evaluating. Every question is built so at least one of its numbers is not a power of the base, so log₂ 6 stays log₂ 6 and the law is the only road to the answer. That is deliberate: a page of log₂ 8 + log₂ 4 can be worked as 3 + 2 = 5 without a law ever being used, which teaches the wrong habit while looking like practice.
How this one works
One question from this sheet, worked through a step at a time.
3 log₁₀ 4
3 log₁₀ 4
k log₁₀ m = log₁₀ (mᵏ)
4³ = 64
3 log₁₀ 4=log₁₀ 64
Both logarithms are to the same base, 10. 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: 4³ = 64. The answer is log₁₀ 64.
