Logarithm Calculator

Logarithm calculator: evaluate log base 10, natural log, or any custom base, with step-by-step conversion to exponential form. Free, runs in your browser.

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What does this generator create?

Logarithm calculator: evaluate log base 10, natural log, or any custom base, with step-by-step conversion to exponential form. Free, runs in your browser.

  • log is the inverse of exponentiation: if 2^10 = 1024, then log2(1024) = 10.
  • Common logs (base 10) count decimal digits; natural logs (base e) appear wherever rates compound continuously.
log base 2 of 100 answer near the bit rounding cutoff
log2(100) = 6.64386, sitting just past the 6-bit cutoff.

When this tool earns its keep

Use it when you need solving for exponents: how many times must the base multiply to reach the value — solving repeatedly or once, with the result ready to copy. Pair it with its related tools below when the output feeds the next step.

Formula & evidence

How this calculator works

Reviewed 2026-09-20 · BoringToolsKit Editorial Team

Formula

log base b of x = ln(x) / ln(b) (change of base), with b set to 10, 2, e or a custom value. Each result is checked by the inverse b^answer.

Site methodology

This page does not cite an external source yet. The link above documents how we build and check calculators; it is our own process, not third-party validation. Treat the inputs and results as editable estimates.

Assumptions to verify

  • Input format is assumed clean; malformed input is reported rather than guessed at.
  • Output is computed client-side and stays in your browser.

Frequently asked questions

Is anything sent to a server?

No. The transform runs in your browser; nothing you paste leaves the page.

What is the honest limitation?

It is mechanical, not judgmental: the tool guarantees format correctness, not fitness for your context — skim the output before shipping it.

Related tools

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Cite this tool

BoringToolsKit. “Logarithm Calculator.” boringtoolskit.com/logarithm-calculator/ (reviewed 2026-09-20). Free to reference in articles, syllabi, and answer posts with a link.

Privacy: Inputs and results stay in this browser. Generated output uses only the fields shown; review and edit it before professional, financial, or legal use.

Worked example: log base 2 of 100, step by step

Set x = 100 and base = 2. The calculator computes log(100)/log(2) using natural logs internally, so the arithmetic you can reproduce by hand is: ln(100) = 4.60517 and ln(2) = 0.69315. Dividing gives 4.60517 / 0.69315 = 6.64386, which is the value shown to four decimals.

Sanity-check it in the other direction: 2^6 = 64 and 2^7 = 128, so the answer must sit between 6 and 7, closer to the top since 100 is past the midpoint of that pair on a log scale. Raising 2 to 6.64386 gives 2^6 x 2^0.64386 = 64 x 1.56267 = 100.00, matching the check line the tool prints. The same inputs with base 10 return exactly 2, because 100 = 10^2; that instant exactness versus the six-decimal base-2 answer is a quick way to see which bases produce clean results and which almost never do.

Where the answers stop being trustworthy

Three inputs produce no number at all: x = 0 or negative (log is undefined there), base = 1 (division by log(1) = 0), and a negative or zero base. The page flags these rather than silently returning NaN. There is also a soft precision limit: for bases very close to 1, say base = 1.001 with x = 2, the result is 693.49, and each extra decimal you type into the base shifts the answer by hundreds. That is mathematically correct but a warning that near-1 bases amplify input noise.

Rounding matters in practice. If you take the base-2 answer 6.64386 and round to 6.64 before exponentiating, you get 2^6.64 = 99.93, about 0.07% off. For order-of-magnitude estimates that is harmless; for bit-length decisions in computing, where log2 answers get rounded UP to whole bits, a value of 6.0001 versus 5.9999 is the difference between 7 and 6 bits allocated. Use the printed check line (base^result = your input) whenever the downstream decision is binary.