Logarithm Calculator

A logarithm answers one question: to what power must you raise the base to get a number? For example, log₁₀(1000) = 3, because 10 × 10 × 10 = 1,000. This logarithm calculator (or log calculator) computes log base 10, natural log (ln, base e), binary log (base 2), or any custom base instantly, using the change-of-base formula where needed, and shows every step.

The definition and the formula

Formally, the logarithm is the inverse of exponentiation. The statement on the left below means exactly the same thing as the statement on the right:

To evaluate a log in any base, the calculator uses the change-of-base formula, which rewrites the log in terms of natural logarithms (or base-10 logs) that a computer can compute directly:

Here x is the value (it must be greater than 0), b is the base (it must be greater than 0 and not equal to 1), and ln is the natural logarithm, log base e, where e ≈ 2.71828.

A worked example

Suppose you want log₂(50): "how many times must I double to reach 50?" Plug into change of base: ln(50) ≈ 3.91202 and ln(2) ≈ 0.69315, so log₂(50) = 3.91202 ÷ 0.69315 ≈ 5.6439. That makes sense: 2⁵ = 32 (too small) and 2⁶ = 64 (too big), so the answer sits between 5 and 6. To check, raise the base back to the result: 2^5.6439 ≈ 50. The reverse calculation returning your original value is the proof that the logarithm is correct.

How to use this logarithm calculator

1. Enter the value (x). This is the number you want the logarithm of. It must be positive, because logarithms of zero or negative numbers are undefined in the real numbers. E-notation such as 2.5e-7 and the letter e both work.
2. Enter the base (b). The default is 10 (the common log). Use the quick buttons for base 2, base 10, or base e, or type any positive base except 1.
3. Read the result instantly. The big number is log_b(x) for your chosen base. Below it you always get the common log, natural log, and binary log of the same value side by side.
4. Follow the steps. The "step by step" card shows ln(x), ln(b), and their quotient, then verifies the answer by raising the base back to the result.

Everything recalculates as you type; there is no submit button to press.

Common log, natural log and binary log

Three bases come up far more than any other, so the calculator always shows all three:

• Common logarithm, log₁₀. Often written just "log". It underpins scientific notation, the decibel scale, earthquake magnitudes, and pH in chemistry. log₁₀(1000) = 3 simply counts the zeros.
• Natural logarithm, ln = log_e. The default of calculus and continuous growth. ln(e) = 1, and the derivative of ln(x) is 1/x.
• Binary logarithm, log₂. The currency of computer science: it counts doublings, bits, and the depth of binary search.

If you need to multiply or raise these results to a power instead, the Exponent Calculator reverses the operation, and a full Scientific Calculator keeps log, ln, and powers on the same keypad.

The logarithm rules

Three identities turn complicated expressions into simple arithmetic. They are the reason logarithms were invented: they convert multiplication into addition.

• Product rule: log_b(x · y) = log_b(x) + log_b(y). Example: log₁₀(100) = log₁₀(10 · 10) = 1 + 1 = 2.
• Quotient rule: log_b(x ÷ y) = log_b(x) − log_b(y). Example: log₂(8/2) = log₂(8) − log₂(2) = 3 − 1 = 2.
• Power rule: log_b(x^k) = k · log_b(x). Example: log₁₀(1000) = log₁₀(10³) = 3 · 1 = 3.

Two anchor identities follow directly from the definition: log_b(1) = 0 (any base to the power 0 is 1) and log_b(b) = 1 (the base to the first power is itself).

Log base 3 of 9, base 2 of 8: change-of-base examples

These are the "log base b of x" problems that show up in homework. Each row applies log_b(x) = ln(x) ÷ ln(b); the clean whole-number results happen when x is an exact power of the base.

Key terms explained

• Base (b): the number being repeatedly multiplied. Logarithms "ask" how many of these multiplications are needed.
• Argument (x): the value inside the log, the target you are trying to reach. It must be positive.
• e: Euler's number, ≈ 2.71828, the base of the natural logarithm and continuous growth.
• Mantissa & characteristic: the fractional and integer parts of a base-10 log. In log₁₀(500) ≈ 2.699, the characteristic 2 tells you the number has 3 digits.
• Antilogarithm: the inverse operation, raising the base to a log value to recover x: b^{(log_b x)} = x.

More worked examples

Example 1: common log of a non-round number. log₁₀(500) = ln(500) ÷ ln(10) = 6.21461 ÷ 2.30259 ≈ 2.6990. The "2" says 500 has three digits; the ".699" places it between 100 (log 2) and 1000 (log 3).

Example 2: natural log in growth. If an investment grows continuously and you want to know the time to double, you solve e^rt = 2, giving t = ln(2) ÷ r. Since ln(2) ≈ 0.6931, at a 5% continuous rate the doubling time is 0.6931 ÷ 0.05 ≈ 13.9 years, the math behind the "rule of 70".

Example 3: a fractional base. log_0.5(8) = ln(8) ÷ ln(0.5) = 2.07944 ÷ (−0.69315) = −3. A base below 1 gives a decreasing log, so the result is negative: you must halve three times to go from 8 down to 1.

What is log(1000000), ln(100) and log2(70)?

These are the exact numbers people most often type. This matrix gives the common log (base 10), natural log (ln), and binary log (base 2) side by side, rounded to five decimal places, so you can read any of them at a glance.

log2(70) step by step

When x is not an exact power of the base, follow the change-of-base steps below. The final row shows the reverse check: raising the base back to the result must return x (here it lands a hair under 70 only because the exponent was rounded to five decimals; the calculator's full-precision check gives 70).

Quick reference table

Some logarithm values are worth recognizing on sight. Notice how log₁₀ of a power of ten is just the number of zeros.

Who this calculator is for

• Students checking algebra, pre-calculus, or calculus homework and learning the change-of-base method (expressing the solution of an exponential equation as a logarithm is a high-school Common Core standard).
• Computer-science learners who need log₂ for algorithm complexity, bits, or tree depth.
• Scientists and engineers working with decibels, pH, earthquake magnitudes, or half-life and decay.
• Finance and data folks using natural logs for continuous growth, log returns, and log-scale charts.
• Anyone who just needs a quick, trustworthy log of a number in a specific base.

Where logarithms show up in the real world

Logarithms compress huge ranges into manageable scales. Earthquake magnitude and acidity (pH) are base-10 logarithmic: the U.S. Geological Survey explains that each whole-number step in magnitude is a tenfold increase in measured amplitude, and each whole pH unit is a tenfold change in acidity. Sound levels in decibels use ten times the base-10 logarithm of an intensity ratio, so every 10 dB is a tenfold change in intensity. In finance, "log returns" use the natural log because they add up neatly over time. In computing, log₂ describes how many comparisons a binary search needs. And the slide rule, the engineer's tool for centuries, worked purely by turning multiplication into the addition of logarithmic distances.

Tips for working with logarithms

• Estimate first. For log₁₀, count digits: a 4-digit number has a log between 3 and 4. This catches typos instantly.
• Memorize log₁₀(2) ≈ 0.301 and log₁₀(3) ≈ 0.477. With the product and power rules you can mentally approximate many other logs.
• Use ln(2) ≈ 0.693 for doubling-time problems; it powers the rule of 70/72.
• Remember the inverse. If a log result looks off, exponentiate it; the base raised to the answer should return your original number.

Solving exponential equations with logarithms

The single most common reason to reach for a logarithm in algebra is to get a variable down out of an exponent. Suppose you need to solve 3^x = 81. Take the log of both sides and apply the power rule: log(3^x) = log(81), so x · log(3) = log(81), which gives x = log(81) ÷ log(3) = 4. The base you choose does not matter (common log, natural log, or log base 3 all return the same x) because the change-of-base ratio cancels out. This is the everyday workhorse use of the change-of-base formula, and it is why "isolate the exponential, then take logs" is a standard recipe in any pre-calculus course.

A subtler case is when the unknown appears in the base instead, as in x³ = 81. That one is a root, not a logarithm, so you would undo it with the root operation rather than a log. The quick test: if the unknown is the power, use a logarithm; if the unknown is the base, use a root or exponent.

Graphing and the shape of a logarithm

Plotting y = log_b(x) reveals why logarithms behave the way they do. Every logarithmic curve passes through the point (1, 0), because log_b(1) = 0 for any base. For a base greater than 1 the curve rises, slowly and ever more gently as x grows, and dives steeply toward negative infinity as x approaches 0 from the right. The vertical line x = 0 is a vertical asymptote the graph never crosses, which is the visual statement that the log of zero is undefined. A larger base produces a flatter curve, and a base between 0 and 1 mirrors the graph downward so the function decreases. Because the logarithm is the inverse of the exponential, its graph is the reflection of y = b^x across the line y = x: the domain and range simply swap, which is why exponentials accept any x but only output positives, while logarithms accept only positive x but output any real number.

A short history: why logarithms were invented

John Napier published the first table of logarithms in 1614, and Henry Briggs reworked them to base 10 a few years later. Their goal was purely practical: before electronic computing, multiplying or dividing large numbers by hand was slow and error-prone. By converting a multiplication into the addition of two table look-ups (the product rule), and a power into a single multiplication (the power rule), logarithms turned hours of arithmetic into minutes. That same idea powered the slide rule, the engineer's constant companion for three centuries, which simply lined up logarithmic distances so that sliding one scale against another physically "added" the logs. Today the arithmetic motivation is gone, but the structural insight, that logarithms convert multiplicative relationships into additive ones, is exactly why they remain everywhere in science, finance, and computing.

Related concepts and calculators

Logarithms sit inside a small family of operations. If your question is slightly different, a sister tool fits better:

• To go the other way and raise a number to a power, use the Exponent Calculator; it is the exact inverse of taking a log.
• For square and other roots, the Square Root Calculator handles the radical case.
• For a full keypad with sin, cos, log and ln buttons, use the Scientific Calculator.
• To write very large or very small results in powers-of-ten form, the Scientific Notation Calculator pairs naturally with base-10 logs.
• For percentages, growth rates, and everyday math, see the Percentage Calculator and Average Calculator.

Logarithms are the tool for solving exponential models. The Exponential Growth Calculator uses the natural log to find how long a quantity takes to grow by a given factor, and the Half-Life Calculator runs the same math in reverse for decay.

Sources & further reading

• NIST Digital Library of Mathematical Functions: §4.2 Logarithm definitions (change of base, equations 4.2.8 and 4.2.9, and log₁₀, equation 4.2.15).
• NIST DLMF: §4.6 Power series for the logarithm.
• Common Core State Standards: HSF-LE.A.4 (express the solution of an exponential model as a logarithm with base 2, 10 or e).
• U.S. Geological Survey: Earthquake magnitude, energy release, and shaking intensity and pH and water (logarithmic scales).
• Napier, J. (1614), Mirifici Logarithmorum Canonis Descriptio, the first published table of logarithms.