FREE ONLINE MATHEMATICS TOOL

Logarithm Calculator — every step, any base

Type a logarithm — natural, common, binary, or a base of your own — and watch the change-of-base formula unfold line by line. Solve logb(x) = y with any one term missing, expand products with the product, quotient and power laws, and see exact answers the moment x is a perfect power of b. Everything re-solves on every keystroke, and nothing leaves the page.

NO SIGN-UP ANY BASE — e · 10 · 2 · CUSTOM LIVE RE-SOLVE EXACT POWER RECOGNITION LOG LAWS INCLUDED
Open the calculator HOW TO USE & FAQ ↓
UPDATED · 2025 — TOOL V1.0
01

The instrument

Three modes, one screen: EVALUATE computes logb(x), SOLVE fills the blank in logb(x) = y, and LOG LAWS expands the three laws with your numbers. The work panel beneath the display rewrites the full proof on every keystroke.

SWS·LOG-01
log
BASE
(
ARGUMENT
)
BASE PRESETS
TRY
THE WORKRE-SOLVES LIVE
log
BASE
(
ARGUMENT
) =
VALUE
LEAVE EXACTLY ONE BOX EMPTY — THAT BOX IS THE UNKNOWN
TRY
THE WORKRE-SOLVES LIVE
log
BASE
(
M
·
N
) ,
M RAISED TO k
THE PRODUCT, QUOTIENT AND POWER LAWS — EXPANDED WITH YOUR NUMBERS
TRY
THE WORKRE-SOLVES LIVE
SWS·LOG-01·RE-SOLVES ON EVERY KEYSTROKE·NOTHING LEAVES THE PAGE
02

How to use it

Ninety seconds from a blank page to working you can copy straight into your notes — the calculator writes the proof, you watch it happen.

STEP 01

Pick your mode

EVALUATE computes logb(x), SOLVE fills the one blank in logb(x) = y, and LOG LAWS expands the product, quotient and power rules with real numbers.

MODE → EVALUATE
STEP 02

Choose a base

Tap the e / 10 / 2 preset chips, or type any base greater than 0 and not equal to 1. The notation adapts itself: ln for base e, log for base 10.

BASE → e · 10 · 2 · CUSTOM
STEP 03

Enter what you know

Decimals, fractions like 3/4, and even the constant e are all accepted. In SOLVE mode, leave exactly one of the three boxes empty — that box is the unknown.

INPUT → 64 — FRACTIONS OK
STEP 04

Watch it solve live

There is no equals button. The screen re-solves on every keystroke, and the work panel beneath rewrites the full proof — domain checks included.

KEYSTROKE → RE-SOLVE
STEP 05

Read the result every way

The big line is the answer; the rows below give the ln, log₁₀ and log₂ conversions, the exponent form bʸ = x, and the exact fraction whenever one exists.

ROWS → ln · log₁₀ · log₂ · bʸ
STEP 06

Follow the work

Change-of-base substitution, exact-power recognition, and the exponentiate-back check — each step numbered, so the whole proof copies into your notes in order.

COPY → STEP 01 … 06
~/worked-exampleLOG LAWS · log₂(8 · 4)
log₂(8 · 4)
├─ PRODUCT LAWlog₂ 8 + log₂ 4
├─ EVALUATE PARTS3 + 2
├─ SUM5
├─ DIRECT CHECKlog₂ 32 = 5 ✓
└─ POWER LAW BONUSlog₂ 8³ = 3 · 3 = 9
CHECK — 2⁵ = 32 = 8 · 4 ✓
03

Frequently asked questions

The questions students actually ask about logarithms — answered the way this tool works them out.

Q·01How do I calculate a logarithm without a calculator?
Rewrite it as an exponent question — logb(x) asks "b raised to what power gives x?" For log₂ 64, factor the argument: 64 = 2 · 2 · 2 · 2 · 2 · 2 = 26, so the answer is 6. When no exact power exists, fall back on the change-of-base formula, logb x = ln x ÷ ln b, and look up the two natural logs.
Q·02How do I solve a log equation with a missing value?
Convert to exponential form. log₃ x = 4 becomes x = 3⁴ = 81; logx 49 = 2 becomes x² = 49, so x = 7. The SOLVE tab fills the blank in any of the three positions — argument, base or value — and then checks the answer back by substitution.
Q·03What is the change-of-base formula and when do I use it?
logb(x) = ln(x) ÷ ln(b) — sometimes written with log₁₀ on top and bottom. It lets you evaluate a logarithm in any base using only natural or common logs, which is exactly how this tool computes custom bases; the substitution is written out in the work panel so you can copy it.
Q·04What is the difference between ln, log and log₂?
Only the base changes. ln is base e ≈ 2.71828, the natural logarithm used for growth, decay and calculus. log on its own means base 10, the common logarithm of the decimal system: log 1000 = 3. log₂ is base 2, the binary logarithm of doubling times, bits and octaves: log₂ 8 = 3. All three are one tap away in the tool.
Q·05What are the product, quotient and power laws of logarithms?
log(M·N) = log M + log N, log(M ÷ N) = log M − log N, and log(Mᵏ) = k·log M. Together they turn multiplication into addition — the property logarithms were invented for. The LOG LAWS tab expands all three with your own numbers and verifies each against the direct computation.
Q·06Why can't I take the log of zero or a negative number?
Because by is strictly positive whenever b > 0 — no exponent ever lands on 0 or below, so the inverse has nothing to return. Real logarithms are defined for positive arguments only; log(−5) exists only in complex numbers, and even there it is multi-valued.
Q·07Why is a base of 1 not allowed?
1y = 1 for every possible y. The exponential collapses everything to the same output, so it is not one-to-one and cannot be inverted — there is no log base 1. The same argument rules out b ≤ 0. A valid base is any positive number other than 1.
Q·08What is an antilog, and how do I compute it?
The antilog is exponentiation — the inverse of taking a logarithm. If logb(x) = y, then antilogb(y) = by = x. To compute one here, open the SOLVE tab and leave the argument box empty: log₃ x = 4 returns x = 81, which is antilog₃ 4.
Q·09Is it free, and does my input leave the page?
Completely free — no sign-up, no locked features, no limits. All arithmetic runs locally in your browser; nothing is recorded or transmitted, and the page keeps working offline once it has loaded.
THE FEED

One worked proof in your inbox, weekly.

New tools, deep dives and honest derivations — no spam, unsubscribe any time.