log(xy) = log x + log y. A product inside becomes a sum outside.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Quotients
log(x / y) = log x − log y, for the same reason.
Powers
log(xⁿ) = n log x. The exponent comes down as a multiplier.
Why they exist
They mirror the exponent laws exactly. A logarithm is an exponent, so its laws are the exponent laws in disguise.
The laws come from the exponent laws
Multiplying powers adds exponents, so log(ab) = log a + log b. Dividing subtracts, so log(a/b) = log a − log b. A power multiplies, so log(aⁿ) = n log a. Each log law is an exponent law read through the inverse.
The power rule is the useful one
log(aⁿ) = n log a brings an exponent down to the front, which is exactly what lets you solve equations with the unknown in the exponent. It is the reason logarithms were invented.
What the laws do not say
log(a + b) is not log a + log b, and log(a)/log(b) is not log(a − b). The laws convert products to sums, never sums to anything. Inventing a law for sums is the standard error here.
Why they mattered historically
Before calculators, logarithms turned multiplication of large numbers into addition, which is far faster by hand. Slide rules and log tables ran on this for three centuries, and it is why navigation and astronomy were possible at all.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Idris Expand a Logarithm
Idris expands log(x²y).
- Step 1
The product law splits it: log(x²) + log y.
Step 4: Your Turn
Practice makes it stick.
The Power
Problem 1 of 2
log(x⁵) equals what multiple of log x?
A Value
Problem 2 of 2
log base 2 of 8. What is it?
Expand and Evaluate
1 of 8
log base 3 of 9. What is it?
2 of 8
log base 5 of 125. What is it?
3 of 8
log base 10 of 1. What is it?
4 of 8
log(x⁷) equals what multiple of log x?
5 of 8
log base 2 of 32. What is it?
6 of 8
Match each expression to its expanded form.
Tap a card on the left to start.
7 of 8
Is log(x + y) the same as log x + log y? 1 for yes, 0 for no.
8 of 8
log base 4 of 16. What is it?
Step 5: Quick Check
Show what you know.
Question 1 of 1
log base 3 of 81. What is it?
What You Learned
- A product inside a logarithm becomes a sum outside it.
- A quotient becomes a difference, and a power becomes a multiplier.
- The laws mirror the exponent laws because a logarithm is an exponent.