Logarithm Calculator

A logarithm answers a single, very practical question: to what power must I raise the base to get this number? Ask "10 to the power of what equals 1,000?" and the answer is 3, because 10 × 10 × 10 = 1,000. Written as a logarithm, that is log₁₀(1000) = 3. This logarithm calculator works that out for any base in an instant - common log (base 10), natural log (ln, base e), binary log (base 2), or a base you type in yourself - and shows the working so you can see exactly where the answer comes from.

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 - logarithms of zero or negative numbers are undefined in the real numbers.
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, the Richter scale, 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.

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).

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.

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.
• Computer-science learners who need log₂ for algorithm complexity, bits, or tree depth.
• Scientists and engineers working with decibels, pH, the Richter scale, 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 (Richter), sound intensity (decibels), and acidity (pH) are all base-10 logarithmic, so each step of 1 means a tenfold change. 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.

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.
• 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.
• For percentages, growth rates, and everyday math, see the Percentage Calculator and Average Calculator.