Logarithm Calculator

What is a Logarithm?

A logarithm answers the question: "To what power must a base be raised to get a given number?" It is the inverse operation of exponentiation.

Types of Logarithms

• Common Log (log₁₀): Used in engineering, pH scale, decibels
• Natural Log (ln): Base e ≈ 2.71828. Essential in calculus and physics
• Binary Log (log₂): Important in computer science and information theory

Logarithm Rules

How the Logarithm Calculator Works

This calculator evaluates log_b(x) for any base you choose. Enter a positive number, pick Common Log (base 10), Natural Log (base e), Binary Log (base 2), or a Custom Base, and click Calculate. The result appears with the formula used — for example, log₁₀(1000) = 3 because 10³ = 1000.

Two inputs are restricted for mathematical reasons: the number must be positive (logarithms of zero and negatives are undefined in real numbers), and a custom base must be positive and not equal to 1.

Worked Examples

Example 1: log₁₀(1000) = 3

Example 2: log₂(32) = 5

Example 3: log₂(1/8) = −3

Logarithms of numbers between 0 and 1 are negative — a useful pattern to recognize.

Example 4: log₁₀(0.001) = −3

Example 5: log₀.₅(8) = −3

Bases between 0 and 1 flip the sign: raising a fraction to a negative power gives a large number.

Example 6: Find the base — in which base is the log of 8 equal to 3?

Example 7: Find the argument — which number has log₁₀(x) = 3?

Example 8: Change of base — log₂(100)

No calculator has a "log₂" button, but every calculator has ln. Convert:

Check: 2^6.6439 ≈ 100. This is exactly what the calculator does internally for custom bases.

Logarithm Rules in Action

The rules table above becomes powerful when applied to real numbers:

• Product: log₁₀(200) = log₁₀(2 × 100) = log₁₀(2) + log₁₀(100) ≈ 0.30103 + 2 = 2.30103
• Quotient: log₁₀(1000 / 10) = log₁₀(1000) − log₁₀(10) = 3 − 1 = 2
• Power: log₁₀(100²) = 2 × log₁₀(100) = 2 × 2 = 4
• Condensing: 2·ln(3) + ln(2) = ln(3²) + ln(2) = ln(9 × 2) = ln(18) ≈ 2.8904

Logarithms Undo Exponentials (and Vice Versa)

Because logs and exponentials are inverse operations, they cancel in both directions:

• log₁₀(10^4.2) = 4.2 — the log strips the base-10 exponential bare.
• 10^(log₁₀(73)) = 73 — the exponential strips the log bare.
• ln(e^(−1.5)) = −1.5 — works for any real exponent, including negatives.

This round-trip property is what makes logarithms the standard tool for solving any equation where the unknown sits in an exponent.

Special Values Worth Memorizing

Why the Argument Must Be Positive (and the Base Cannot Be 1)

Two inputs are off-limits, and both restrictions come straight from the definition:

• Argument x > 0: b^y is always positive when b > 0, so no real exponent y can produce 0 or a negative number. That is why log(0) and log(−5) are undefined.
• Base b > 0, b ≠ 1: 1^y = 1 for every y, so a base-1 logarithm could never give a unique answer — log₁(5) would have to be every number at once.

Common Log vs. Natural Log: Which Should You Use?

Notation warning: a bare "log" usually means base 10 in engineering and school math, but base e in many university textbooks and programming languages. When the base matters, write it explicitly.

Where e Comes From: A Compound-Interest Story

Euler's number e ≈ 2.718281828 emerges from a simple question: what happens to $1000 invested at 100% annual interest as compounding gets more frequent? The growth factor is (1 + 1/m)^m, where m is the number of compounding periods:

No matter how often you compound, the factor never exceeds e — it is the speed limit of 100% growth. That is why continuous growth uses base e, and why the natural logarithm ln is the natural choice for anything that grows or decays smoothly.

Estimating Logarithms Mentally: Characteristic and Mantissa

Before calculators, people split every common log into two parts: the characteristic (the integer part, read straight off the number of digits) and the mantissa (the decimal part, looked up in a table).

• Characteristic = 3 — 7320 has 4 digits, so its log is between 3 and 4.
• Mantissa ≈ 0.8645 — depends only on the digit sequence 7.32, not on where the decimal point sits.

That separation is why old log tables only listed mantissas: log₁₀(732), log₁₀(73.2), and log₁₀(7320) all share the mantissa 0.8645 and differ only in the characteristic (2, 1, and 3). Slide rules worked on the same principle — the mantissa is the distance along the rule.

Real-World Applications

• pH scale (chemistry): pH = −log₁₀[H⁺]. A hydrogen concentration of 10⁻⁷ mol/L gives pH 7 (neutral); each pH unit is a 10× change in acidity.
• Richter scale (seismology): each whole-number step means 10 times the wave amplitude — a magnitude 6 quake shakes 10× harder than a magnitude 5.
• Decibels (acoustics): dB = 10·log₁₀(I/I₀). Doubling sound intensity adds about 3 dB (10·log₁₀(2) ≈ 3.01); 10× the intensity adds exactly 10 dB.
• Bits (information theory): choosing among 8 equally likely outcomes carries log₂(8) = 3 bits of information.
• Log scales (data visualization): when data spans many orders of magnitude — earthquake energies, stellar brightness — a logarithmic axis keeps everything visible.
• Algorithm analysis (computer science): binary search halves the problem each step, so its cost grows like log₂(n).

A Short History of Logarithms

Logarithms were invented to turn the hardest arithmetic of the 1600s — multiplying and dividing huge numbers for astronomy and navigation — into simple addition. Scottish mathematician John Napier published the first logarithm tables in 1614 after twenty years of work; Henry Briggs then rebuilt them around base 10, the version scientists used for the next 350 years.

The word itself comes from Greek logos (ratio) and arithmos (number): "ratio-numbers." Edmund Gunter (1620) and William Oughtred (1622) engraved logarithms onto sliding rulers, creating the slide rule — the pocket calculator of engineers, scientists, and Apollo astronauts until electronic calculators arrived in the 1970s.

Solving Logarithmic Equations

Three strategies cover most equations: convert to exponential form, condense with log rules, and always check the domain at the end.

Equation 1: log₂(x) = 5

Equation 2: log(x) + log(x − 3) = 1

1. Condense: log(x(x − 3)) = 1
2. Exponentiate: x(x − 3) = 10¹ = 10
3. Solve: x² − 3x − 10 = 0 → x = 5 or x = −2
4. Check the domain: x = −2 makes log(x) undefined, so reject it. Answer: x = 5.

Equation 3: ln(x) = 2.5

Reading the Graph of y = log_b(x)

• Always passes through (1, 0) — because log_b(1) = 0 in every base.
• Vertical asymptote at x = 0 — the curve dives toward −∞ as x approaches 0 from the right.
• Rises when b > 1 (log₁₀, ln, log₂) and falls when 0 < b < 1 (log₀.₅).
• Domain is x > 0 only — there is no part of the graph left of the y-axis.

Common Mistakes to Avoid

• log(x + y) ≠ log(x) + log(y). The product rule works on multiplication inside the log, not addition. log(100 + 100) = log(200) ≈ 2.30, but log(100) + log(100) = 4.
• log(x) / log(y) ≠ log(x / y). Division of logs is the change-of-base formula in disguise: log(x)/log(y) = log_y(x).
• Forgetting the domain when solving equations. Any candidate that makes an argument zero or negative must be rejected, as in Equation 2 above.
• Assuming "log" always means base 10. Check the field — mathematicians and many programming languages mean base e.
• Trying the log of zero or a negative. In real numbers these are undefined; the calculator will refuse them.