Logarithm Rules Discovery

Algebra 2 · Axiom Academy

One quiet idea hides inside every logarithm rule: it turns the harder operation into an easier one. Spot the pattern for yourself and you'll rediscover — and two more just like it. What multiplying does to a logarithm Logarithms have a hidden talent. Multiply two numbers together and their logarithm doesn't multiply — it does something far simpler. Let's catch it in the act with a pair of friendly numbers: 100 and 1000 . Watch computed two ways at once. Below, the two logarithms and lay end to end. Above and below the same ruler, the length they reach and the log of the product 100000 race to the exact same mark. Two routes, one destination: adding the logs gets you to exactly where taking the log of the product does. The logarithm of a product is the sum of the logarithms. Multiplication on the inside becomes addition on the outside. If multiplying adds, what does dividing do? Same game, one operation over. Choose how many tens go into a and into b , then read the ruler: walk right by , then back left by , and you land exactly on . Try to make the pattern fail. Dividing the numbers subtracts their logarithms — the mirror image of the product rule. The logarithm of a quotient is the difference of the logarithms. Division on the inside becomes subtraction on the outside. And what about raising to a power?

This is the written version of the interactive lesson above. See the full Algebra 2 course.