Logarithms
A logarithm answers one question: what exponent do I need?
Exponents run backwards
Two statements say exactly the same thing. Move the sliders and watch them stay in step.
Facts worth memorizing
The base b must be positive and not equal to 1. The input x must be positive, because no power of a positive base is zero or negative.
| logb(1) = 0 | because b0 = 1 |
| logb(b) = 1 | because b1 = b |
| logb(bn) = n | the log undoes the power |
| blogb(x) = x | the power undoes the log |
| log(x) means log10(x) | ln(x) means loge(x), with e ≈ 2.718 |
| logb(x) = log(x)log(b) | change of base: for example, log2(3) = log(3)log(2) ≈ 1.585. ln works too: ln(3)ln(2). |
Transform y = logb(x − h) + k
The orange curve is the exponential inverse. Both curves mirror each other across the dashed line y = x.
logarithm inverse exponential y = x
The three laws, tested with your numbers
Pick any values. The two sides of each law always match.
When to use which
Product and quotient laws turn multiplying and dividing into adding and subtracting. The power law pulls an exponent out front, which is how you solve for a variable stuck in an exponent: take the log of both sides. Change of base lets a calculator, which only has log and ln, handle any base.
Players and progress are saved in this browser only.