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Algebra

Logarithm (any base)

logₐ(x). Free online Logarithm (any base). Calculate logarithm (any base) online — fast, accurate, mobile-friendly, no signup needed.

log
2

Derivation

  1. ├── 01Givenx = 100, a = 10
  2. ├── 02Formulaln(t) / ln(a)
  3. ├── 03Substituteln(t) / ln(10)
  4. └── 04Compute log2
Did you know?

Søren Sørensen invented the pH scale in 1909 while studying protein chemistry at the Carlsberg brewery — "p" stands for "potenz" (power) and "H" for hydrogen.

§01What is

Understanding the Logarithm (any base)

The Logarithm (any base) computes log from 2 inputs: x, base a. logₐ(x).

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Logarithm (any base) sits in that toolkit — it logₐ(x). Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

ln(t) / ln(a)

Where

x
x
a
base a
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: x = 100, base a = 10.

  1. 01Start by noting the input — x: 100.
  2. 02Start by noting the input — base a: 10.
  3. 03Substitute these values into the formula: ln(t) / ln(a)
  4. 04Compute log: the calculator returns 2.
  5. 05Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Logarithm (any base) Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

x halved

x = 50 (from 100)

Keep every other input at its default and halve the x. See how log responds.

  1. 01New x: 50
  2. 02Baseline log: 2
  3. 03New log: 1.69897
  4. 04log decreases by 15.1% → use this sensitivity to plan for real-world variation.
02 · PATTERN

x doubled

x = 200 (from 100)

Keep every other input at its default and double the x. See how log responds.

  1. 01New x: 200
  2. 02Baseline log: 2
  3. 03New log: 2.30103
  4. 04log increases by 15.1% → use this sensitivity to plan for real-world variation.
03 · PATTERN

base a halved

a = 5 (from 10)

Keep every other input at its default and halve the base a. See how log responds.

  1. 01New base a: 5
  2. 02Baseline log: 2
  3. 03New log: 2.86135
  4. 04log increases by 43.1% → use this sensitivity to plan for real-world variation.
04 · PATTERN

base a doubled

a = 20 (from 10)

Keep every other input at its default and double the base a. See how log responds.

  1. 01New base a: 20
  2. 02Baseline log: 2
  3. 03New log: 1.53724
  4. 04log decreases by 23.1% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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