Logarithm Calculator
Calculate logarithms with any custom base. Shows change of base formula, common log, natural log, and binary log conversions.
Log Calculator
Results update instantly as you type
Enter Values
Change of Base
log_2(8) = log₁₀(8) / log₁₀(2) = 0.903089987 / 0.3010299957 = 3
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log_2(8)
3
Logarithmic Curve f(x) = log2(x)
Step-by-Step Calculation
Exponential Form
2^3 = 8
Change of Base Formula
log_2(8) = log₁₀(8) / log₁₀(2) = 0.903089987 / 0.3010299957 = 3
Use this formula when your calculator only has log₁₀ or ln buttons.
log₁₀
0.903089987
ln (base e)
2.079441542
log₂
3
The Formula
A logarithm answers the question: "to what exponent must the base be raised to produce this number?" The change of base formula allows calculating logarithms in any base using the log₁₀ or ln buttons found on standard calculators.
Variable Definitions
Base
The base of the logarithm. Must be positive and not equal to 1. Common bases: 10 (common log), e (natural log), 2 (binary log).
Argument
The number whose logarithm is being taken. Must be positive.
Result
The exponent such that b^y = x. For example, log₂(8) = 3 because 2³ = 8.
How to Use This Calculator
- 1
Enter the base of the logarithm (must be > 0, and cannot be 1).
- 2
Enter the number to take the log of (must be > 0).
- 3
View the logarithm result for your custom base.
- 4
The calculator also shows common (log₁₀), natural (ln), and binary (log₂) logs of the same number for comparison.
- 5
The change of base formula shows step-by-step how the result was computed using base-10 logs.
Logarithms are the inverse of exponentiation: log_b(x) asks what power of b equals x
Understanding the Concept
Logarithms are the inverse of exponentiation. Just as division undoes multiplication, logarithms undo exponentiation. The fundamental relationship is: log_b(x) = y means b^y = x. For example, log₂(8) = 3 because 2³ = 8. Logarithms are essential in chemistry (pH = -log₁₀[H⁺], where each unit represents a 10x change in acidity), physics (decibel and Richter scales are logarithmic), biology (population growth models), computer science (binary search, algorithm complexity analysis where log₂ n appears constantly), and finance (logarithmic returns for compound interest). The change of base formula, log_b(x) = log_c(x) / log_c(b), is crucial because most calculators only provide log₁₀ (common log) and ln (natural log) buttons. For example, to compute log₂(100) on a standard calculator: log₁₀(100) / log₁₀(2) = 2 / 0.3010 = 6.644. The calculator shows this step-by-step. A key property: logarithms turn multiplication into addition (log(xy) = log(x) + log(y)), which is how slide rules worked before electronic calculators.
Frequently Asked Questions
Sources & References
Related Calculators
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