Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 logarithms? Then this Algebra 2 lesson on Logarithms: The Inverse of Exponents explains each idea step by step, with worked examples.
Quick answer: A logarithm is the exponent that a base must be raised to in order to produce a given number. For example, log base 2 of 32 equals 5 because 2 raised to the 5th power is 32. Logarithms are the inverse of exponential functions, and their graphs are mirror images across the line y = x.
Standards: F-LE.A.4, F-BF.B.4a, F-IF.C.7e · Level: Grade 11 Algebra 2
What is a logarithm?
A logarithm answers the question “which exponent do I need?” Because 25 = 32, the logarithm of 32 in base 2 is 5.
For b > 0, b ≠ 1 and x > 0: logb x = y exactly when by = x. A log with no written base means base 10 (log), and ln means base e ≈ 2.718.
What does the graph of a logarithmic function look like?
The graph of y = log2 x passes through (1, 0) because b0 = 1 for every base. It rises slowly, and the y-axis is a vertical asymptote. The domain is x > 0 and the range is all real numbers.
Since logarithms undo exponentials, the two graphs are mirror images over the line y = x.
Which properties make logarithms useful?
- Product: logb(MN) = logb M + logb N
- Quotient: logb(MN) = logb M − logb N
- Power: logb(Mp) = p · logb M
- Change of base: logb x = ln xln b
How do you expand a logarithm?
- Write the argument as a product, quotient and powers.
- Split products into sums and quotients into differences.
- Move each exponent in front as a multiplier.
Expand log5(25a3c).
log5 25 + log5 a3 − log5 c = 2 + 3 log5 a − log5 c.
logb(M + N) is not logb M + logb N. There is no rule for the log of a sum.
Where are logarithms used?
Logarithmic scales squeeze huge ranges into small numbers: pH = −log[H+], sound level L = 10 log(I/I0) in decibels, and earthquake magnitude. Adding 1 to pH means ten times fewer hydrogen ions.
Say it as a question: “log3 81 means 3 to what power gives 81?” The answer, 4, is the log.
- logb x = y means by = x, with b > 0, b ≠ 1 and x > 0.
- The graph of y = logb x passes through (1, 0) and has the y-axis as a vertical asymptote.
- Product, quotient and power rules turn multiplication into addition, division into subtraction and exponents into multipliers.
- Change of base: logb x = ln x ÷ ln b, which lets a calculator find any logarithm.
- log means base 10 and ln means base e ≈ 2.718.
Logarithms: The Inverse of Exponents: frequently asked questions
What is the difference between log and ln?
The notation log means base 10, and ln means base e, where e is about 2.718. Both follow the same properties. Use log for powers of 10 and ln when e appears, as in continuous growth.
Why can you not take the log of a negative number?
A positive base raised to any real power always gives a positive result. No exponent can produce a negative number or zero, so the argument of a logarithm must be greater than 0.
How do you find log base 3 of 50 on a calculator?
Use change of base: log3 50 = ln 50 ÷ ln 3 ≈ 3.912 ÷ 1.099 ≈ 3.56. Calculators have only log and ln keys, so every other base is converted this way.
Why does every logarithmic graph pass through (1, 0)?
Any base raised to the power 0 equals 1, so logb 1 = 0 for every valid base b. That is why the x-intercept of y = logb x is always 1.
What does log(AB) = log A + log B really say?
Multiplying two numbers adds their exponents when written with the same base. Since a logarithm is an exponent, the log of a product is the sum of the logs. This is why logs once replaced slide-rule multiplication.