log₂8=3, log₂16=4, log₂128=7. Look closely at these three numbers and you'll notice 3+4=7. And since 8×16=128, this isn't a coincidence. log₂(8×16) = log₂8 + log₂16 holds exactly. This is the most famous property of logarithms: the rule that "multiplication turns into addition."

Why does this happen? A logarithm is really a calculation that asks "what power do I need to raise this to, to get this number?" log₂8=3 means "multiplying 2 by itself 3 times gives 8," and log₂16=4 means "multiplying 2 by itself 4 times gives 16." So 8×16 is (2×2×2)×(2×2×2×2) — you're joining together "2 multiplied by itself 3 times" and "2 multiplied by itself 4 times." By the exact same principle behind the law of exponents aᵐ×aⁿ=am+n, you get 128, which is 2 multiplied by itself a total of 3+4=7 times.

8 = 2³ 16 = 2⁴ 8 × 16 = 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128 log₂8 + log₂16 = 3 + 4 = 7 = log₂128
Exponents adding inside a multiplication, and logarithm values adding together, are two ways of expressing the same fact

Division becomes subtraction by this exact same principle. loga(M÷N) = logaM − logaN, which corresponds to the law of exponents xm÷xn=xm−n. For example, log₂(16÷8) = log₂2 = 1, and log₂16−log₂8 = 4−3 = 1 gives the exact same answer.

Powers are a bit more interesting. loga(Mk) = k×logaM — since Mk is M multiplied by itself k times, you can think of this as applying the multiplication rule k−1 times over. log₂(8²) = log₂64 = 6, and 2×log₂8 = 2×3 = 6 give the exact same answer. This property, where the exponent jumps out in front of the logarithm and becomes multiplication, is frequently used to simplify complicated exponent calculations using logarithms.

Why do these properties matter so much? Before computers existed, mathematicians and scientists used logarithm tables to turn the multiplication of very large numbers into much easier addition. This is exactly why logarithms are still used today whenever numbers span an enormous range — like the intensity of an earthquake (the Richter scale) or the loudness of a sound (decibels) — because they turn multiplication and division relationships into addition and subtraction, which is far easier to handle.

On our activity page, you can change the base a and the values M and N with sliders, and verify like a calculator whether the values before applying a logarithm (multiplication, division, a power) and after applying one (addition, subtraction, multiplication) always come out equal. Try many different combinations of numbers and confirm for yourself that "yes, it really always holds."