What is a logarithm?
A logarithm is the power you raise a base to, to get a number: y = logb(x) means the same as by = x. So log₁₀(1,000) is 3, because 10³ is 1,000. The two you’ll meet most are the common log, base 10, often written just “log”, and the natural logarithm (ln), base e. The base must be above 0 and not 1, and the number must be above 0.
Source OpenStax College Algebra 2e · 6.3 Logarithmic Functions: definition, common and natural logarithms ↗
How to calculate a logarithm with any base
Here it is with your numbers:
- log₅(36) = ln(36) ÷ ln(5)
- log₅(36) = 3.583519 ÷ 1.609438
- log₅(36) = 2.2266
It means you raise 5 to the power 2.2266 (rounded) to get 36.
This is the change-of-base formula: logb(x) = ln(x) ÷ ln(b). It works with log (base 10) in place of ln too, which is how you find any logarithm on a calculator that only has log and ln buttons.
Source OpenStax College Algebra 2e · 6.5 Logarithmic Properties: the change-of-base formula ↗
Common log and natural log table
Tap a number to see the steps.
| x | log(x) | ln(x) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.301 | 0.6931 |
| 3 | 0.4771 | 1.0986 |
| 4 | 0.6021 | 1.3863 |
| 5 | 0.699 | 1.6094 |
| 6 | 0.7782 | 1.7918 |
| 7 | 0.8451 | 1.9459 |
| 8 | 0.9031 | 2.0794 |
| 9 | 0.9542 | 2.1972 |
| 10 | 1 | 2.3026 |
| 20 | 1.301 | 2.9957 |
| 50 | 1.699 | 3.912 |
| 100 | 2 | 4.6052 |
| 1,000 | 3 | 6.9078 |
FAQs
What is a logarithm?
A logarithm is an exponent. logb(x) is the power you raise b to, to get x: log₂(8) is 3, because 2³ is 8. log written without a base usually means base 10, the common log.
Source OpenStax College Algebra 2e · 6.3 Logarithmic Functions: definition, common and natural logarithms ↗
What is natural log?
The natural log, ln, is the logarithm with base e, an irrational number that is 2.718282 to six decimal places. So ln(x) is the power you raise e to, to get x, and ln(1) is 0.
Source OpenStax College Algebra 2e · 6.1 Exponential Functions: the number e ↗
How to evaluate logarithms?
Ask what power of the base gives the number. For log₃(1/27): 3⁻³ is 1/27, so log₃(1/27) is −3. When it isn’t a whole power, use the change-of-base formula below, or this calculator.
Source OpenStax College Algebra 2e · 6.3 Logarithmic Functions: definition, common and natural logarithms ↗
How to use logarithms in calculator?
Most calculators only have log (base 10) and ln (base e). For any other base, divide: logb(x) = log(x) ÷ log(b), or ln(x) ÷ ln(b), which gives the same answer. log₅(36) is log(36) ÷ log(5), which is 2.2266.
Source OpenStax College Algebra 2e · 6.5 Logarithmic Properties: the change-of-base formula ↗
Can the base of a logarithm be negative?
No. The base has to be a positive number other than 1, and you can only take the log of a number above 0.
Source OpenStax College Algebra 2e · 6.3 Logarithmic Functions: definition, common and natural logarithms ↗
Sources
- OpenStax College Algebra 2e: 6.3 Logarithmic Functions: definition, common and natural logarithms
- OpenStax College Algebra 2e: 6.1 Exponential Functions: the number e
- OpenStax College Algebra 2e: 6.5 Logarithmic Properties: the change-of-base formula