Logarithm Calculator
Calculate logarithms
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Need a fast answer on calculating logarithms? Logarithm Calculator answers with the steps beside it. Type Number and Base into Logarithm; each keystroke re-runs the math. Nothing in Logarithm is a black box. Logarithm replaces guesswork. Nothing is uploaded while Logarithm works, and the figures stay put. Logarithm sits in the Math section next to free online logarithm calculator and free logarithm calculator. Run Logarithm twice to find the driving input. Start with Logarithm; the first number takes seconds. Logarithm is linked from the Math hub page.
What the defaults give
log₁₀(100) = 2.0000, since 10 raised to 2 is 100.
The engine computes it as ln(100) ÷ ln(10) = 4.6052 ÷ 2.3026.
The formula
log_b(x) = ln(x) ÷ ln(b)
Equivalently: y = log_b(x) means bʸ = x
A logarithm answers the question "what power do I raise the base to, to get this number?" Any base works, and the change-of-base formula converts between them — which is how a calculator that only knows natural logs can return base 2, base 7 or base 1.5 answers.
Worked examples
log₁₀(100) → 2.0000
log₁₀(1,000) → 3.0000
log₂(8) → 3.0000
log₂(1,024) → 10.0000
log₅(25) → 2.0000
log₁₀(0.01) → −2.0000
The rules worth memorising
log(xy) = log x + log y turns multiplication into addition, which is why logarithms were invented and why slide rules worked. log(x/y) = log x − log y does the same for division, and log(xⁿ) = n log x pulls a power down as a multiplier. Those three rules are the reason logs appear wherever quantities multiply or grow exponentially.
Two fixed points follow: log(1) = 0 in every base, because any number to the power zero is one, and log(base) = 1. A logarithm of zero or a negative number has no real answer, since no power of a positive base gives zero or a negative result.
Which base to use
Base 10 suits orders of magnitude: a pH of 3 is ten times more acidic than a pH of 4, and a Richter magnitude of 6 releases about ten times the energy of magnitude 5. Base 2 suits computing, where doubling is the native operation — a 20-bit address space holds 2²⁰, or just over a million, addresses. Base e (2.718) suits continuous growth and decay, and is the natural log because the derivative of eˣ is itself. In acoustics a 10 dB rise is ten times the intensity, so a 30 dB difference means a factor of a thousand.
The choice of base changes the number, not the relationship. log₂(100) is 6.64 while log₁₀(100) is 2, but both describe the same point, and converting between them is a single constant: log₂(x) = 3.3219 × log₁₀(x).
Frequently Asked Questions
What is log base 10 of 100?
2, because 10² = 100. The calculator reaches it through ln(100)/ln(10) = 4.6052/2.3026.
How do I change the base of a logarithm?
Divide by the log of the new base: log_b(x) = ln(x) ÷ ln(b). For example log₂(100) is 4.6052 ÷ 0.6931 = 6.6439, which is also 3.3219 times log₁₀(100).
Why is log(1) always zero?
Because any base raised to the power zero is 1. Equivalently, ln(1) is 0 and dividing by ln(b) leaves 0.
Can I take the log of a negative number?
Not as a real number. No positive base raised to any power gives a negative result; the complex answer needs a different treatment.
When should I use base 2 rather than base 10?
Base 2 for computing and doubling, such as address space or algorithmic complexity. Base 10 for orders of magnitude like pH or decibels.
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