Log and Exponent Rules
The two sets of laws are the same statements read in opposite directions. Seeing that makes both far easier to remember.
Last updated: August 2026 · Written and verified by Akash Pandey
They are inverses
Everything below follows from one relationship:
log_b(x) = n ⟺ bⁿ = x
Each index law has a mirror-image log law, because a logarithm converts one operation into the operation one level simpler. Multiplication becomes addition, powers become multiplication.
| Exponent law | Corresponding log law | What changed |
|---|---|---|
| bᵐ × bⁿ = bᵐ⁺ⁿ | log(xy) = log x + log y | Multiply → add |
| bᵐ ÷ bⁿ = bᵐ⁻ⁿ | log(x/y) = log x − log y | Divide → subtract |
| (bᵐ)ⁿ = bᵐⁿ | log(xⁿ) = n · log x | Power → multiply |
| b^(1/n) = ⁿ√b | log(ⁿ√x) = (log x)/n | Root → divide |
| b⁰ = 1 | log_b(1) = 0 | Identity |
| b¹ = b | log_b(b) = 1 | Identity |
The index laws in full
| Rule | Example |
|---|---|
| bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| (bᵐ)ⁿ = bᵐⁿ | (5²)³ = 5⁶ = 15,625 |
| (ab)ⁿ = aⁿ · bⁿ | (2×5)³ = 8 × 125 = 1,000 |
| (a/b)ⁿ = aⁿ / bⁿ | (3/4)² = 9/16 |
| b⁰ = 1 (b ≠ 0) | 99⁰ = 1 |
| b⁻ⁿ = 1/bⁿ | 4⁻² = 1/16 = 0.0625 |
| b^(m/n) = ⁿ√(bᵐ) | 16^(3/4) = 8 |
The log laws in full
| Rule | Example (base 10) |
|---|---|
| log(xy) = log x + log y | log(50) = log(5) + log(10) ≈ 1.69897 |
| log(x/y) = log x − log y | log(0.5) = log(5) − log(10) ≈ −0.30103 |
| log(xⁿ) = n · log x | log(8) = 3 · log(2) ≈ 0.90309 |
| log(ⁿ√x) = (log x)/n | log(√1000) = 3/2 = 1.5 |
| log_b(x) = log(x)/log(b) | log₂(32) = log(32)/log(2) = 5 |
| log_b(1) = 0 | log(1) = 0 |
| log_b(b) = 1 | log(10) = 1 |
| b^(log_b x) = x | 10^log(7) = 7 |
Change of base
Any logarithm can be rewritten in any other base by a single division. This matters because most calculators offer only base 10 and base e directly.
log_b(x) = log_a(x) ÷ log_a(b)
- The intermediate base a can be anything, as long as it is the same on top and bottom.
- In practice use base 10 or base e, since those are the ones you have.
- Example:
log₇(200) = ln(200) ÷ ln(7) ≈ 5.298317 ÷ 1.945910 ≈ 2.722706.
This calculator accepts a second argument directly - log(200, 7) - and performs the change of base internally, so you rarely need to do it by hand.
Frequently Asked Questions
Is log(x + y) the same as log x + log y?
No, and this is the most common mistake with logarithms. log x + log y equals log(xy), not log(x+y). A logarithm of a sum has no simplification.
How do I change the base of a logarithm?
Divide by the log of the new base: log_b(x) = log(x) ÷ log(b). Any consistent intermediate base works.
Why does log(1) equal 0 in every base?
Because any base raised to the power 0 gives 1. The logarithm asks which power produces 1, and the answer is always 0.
What is b to the power of log base b of x?
It is x. Exponentiation and logarithms in the same base undo each other exactly.
Why do the exponent and log laws mirror each other?
Because a logarithm is the inverse of exponentiation. Each log law is an index law read backwards, which is why multiplication on one side always corresponds to addition on the other.