Logarithm Calculator
Calculate logarithms for any base
What is the Logarithm Calculator?
A logarithm answers the question: "What exponent do I need to raise this base to in order to get this number?" That's it. Log base 10 of 1000 = 3 because 10³ = 1000. CalciHub's logarithm calculator handles any base — including the natural log (base e) and log base 2, which shows up in computer science. Enter your base and your number, and it returns the result along with the equivalent exponential form.
How Does It Work?
A logarithm is the inverse of exponentiation. If bˣ = y, then log_b(y) = x. The most common bases are 10 (common log), e ≈ 2.718 (natural log, written ln), and 2 (binary log, used in computing). Every log calculation can be rewritten as an exponent, and that equivalence is the key to understanding what the number actually means.
log_b(y) = x means b^x = y Change of base: log_b(y) = ln(y) / ln(b)
• b = the base of the logarithm
• y = the number you're taking the log of (must be positive)
• x = the result (the exponent)
• ln = natural logarithm (base e)
• e ≈ 2.71828 = Euler's number
How to Use CalciHub's Logarithm Calculator
1. Select or enter your base (10, e, 2, or a custom number).
2. Enter the number you want to find the log of.
3. Click Calculate — the tool returns the logarithm value.
4. The result also shows the equivalent exponential statement.
Tip: The argument (the number you're logging) must always be positive — logarithms of zero or negative numbers are undefined in real numbers.
A Quick Example
Find log base 2 of 64. Enter base = 2, number = 64, then click Calculate. The answer is 6, because 2⁶ = 64. Now try the natural log: ln(20) with e as the base. The result is approximately 2.996, which means e^2.996 ≈ 20. This comes up in continuous growth and decay problems. The change-of-base formula confirms it: ln(20)/ln(e) = ln(20)/1 = ln(20).
Log base 2 is worth getting comfortable with — every time you double the input, the log increases by exactly 1, which is why it maps so naturally to binary data and algorithm complexity.
Frequently Asked Questions
Log usually means log base 10 (common logarithm). Ln means log base e (natural logarithm). In most math textbooks, log without a base specified means base 10. In higher math and calculus, it often means natural log. Context matters.