Log Calculator

What is a log calculator?

A log calculator finds the logarithm of a number: the power you must raise a base to in order to get that number. Enter a number, choose a base (10, e, 2 or any custom base) and the calculator gives the answer with steps using the change of base formula. It also shows the common log, natural log (ln), binary log, the characteristic and mantissa, and it works in reverse as an antilog calculator. It is free and handy for maths, chemistry (pH), physics (decibels), finance and CA Foundation or class 11 practice.

What is a logarithm?

A logarithm answers the question “what power?”. Since 10³ = 1,000, the log of 1,000 to base 10 is 3, written log₁₀(1000) = 3. In general, if b^y = x, then log_b(x) = y. Logarithms are the inverse of exponents, which is why the exponent calculator and the log calculator undo each other. The base b must be positive and not equal to 1, and x must be positive: there is no real log of 0 or of a negative number.

How to use the log calculator

Choose Logarithm (or Antilog to go the other way).
Enter the number x. For antilog, enter the log value y.
Select the base: 10 for the common log, e for the natural log ln, 2 for the binary log, or Custom base and type any base such as 3, 5 or 1.5.
Read the answer and the step-by-step change of base working.

Log formula and change of base

log_b(x) = y ⇔ b^y = x | log_b(x) = ln(x) ÷ ln(b) = log₁₀(x) ÷ log₁₀(b)

Most calculators only have log (base 10) and ln (base e) keys. To find a log with a different base, divide: log₂(64) = log(64) ÷ log(2) = 1.80618 ÷ 0.30103 = 6. This is exactly what the calculator does for a custom base, and it shows both numbers in the steps.

Types of logarithms

Name	Base	Written	Used in
Common log	10	log x or log₁₀ x	pH, decibels, Richter scale, log tables
Natural log	e ≈ 2.71828	ln x	Calculus, growth and decay, continuous interest
Binary log	2	log₂ x or lb x	Computer science, bits, algorithms
Any base	b > 0, b ≠ 1	log_b x	Algebra, solving exponential equations

Log vs ln on a calculator

The log key on a calculator means base 10; the ln key means base e. They differ by a constant factor: ln x = 2.302585 × log x. So log(100) = 2 but ln(100) ≈ 4.60517.

Logarithm rules

Product: log_b(xy) = log_bx + log_by
Quotient: log_b(x/y) = log_bx − log_by
Power: log_b(xⁿ) = n × log_bx
Special values: log_b1 = 0, log_bb = 1, b^log_bx = x
Change of base: log_bx = log_kx ÷ log_kb for any valid k

These rules turn multiplication into addition, which is why log tables were used for centuries before electronic calculators.

Worked examples

log₁₀(1000) = 3, because 10³ = 1,000 (the default above).
log₂(8) = 3, because 2³ = 8. log₂(1024) = 10.
ln(e²) = 2, and ln(10) ≈ 2.302585.
log₃(81) = 4, because 3⁴ = 81. Choose Custom base 3 and x = 81 to check.
log₁₀(0.01) = −2. Numbers between 0 and 1 have negative logs.
log₁₀(2) ≈ 0.30103, a value worth remembering for quick mental estimates.

Characteristic and mantissa (log table method)

In the log table method taught in Indian schools, log₁₀ x is split into an integer part, the characteristic, and a decimal part, the mantissa. For log(345.6) = 2.53857, the characteristic is 2 (one less than the number of digits before the decimal point) and the mantissa, 0.53857, comes from the table. The calculator shows both for any number, so you can check log table work.

How to find the antilog

The antilog is the inverse of the log: antilog_b(y) = b^y. The common antilog of 2.5 is 10^2.5 ≈ 316.228, and the natural antilog of 1 is e ≈ 2.71828. Select “Antilog” above, enter y and choose the base. On a scientific calculator, antilog is usually SHIFT + log (10ˣ) or SHIFT + ln (eˣ).

Where logarithms are used

pH in chemistry: pH = −log₁₀[H⁺]. A hydrogen-ion concentration of 0.0001 mol/L gives pH = 4.
Sound: decibels = 10 × log₁₀(I/I₀); every +10 dB is ten times more intense.
Finance: the time to grow money at rate r is n = log(A/P) ÷ log(1 + r). At 8% a year, money doubles in log 2 ÷ log 1.08 ≈ 9 years (compare the compound interest calculator).
Computer science: binary search on n items takes about log₂ n steps; 1,000,000 items need only about 20.

How to calculate log in a normal calculator

A basic calculator has no log key, but you can estimate: count the digits for the characteristic, then use log 2 ≈ 0.301, log 3 ≈ 0.477 and log 7 ≈ 0.845 with the product rule. log(6) = log 2 + log 3 ≈ 0.778. For exact values use this log calculator or a scientific calculator.

Log calculator meaning and name

"Log" is short for logarithm, a name John Napier coined in 1614 from the Greek words for "ratio" and "number". The log calculator meaning is simple: log_b(x) answers "what power of b gives x?" So log₁₀(1000) = 3 because 10³ = 1000.

Logarithm formula calculator: log form and exponent form

The logarithm formula connects two forms: log form log_b(x) = y is the same as exponent form b^y = x. A log form calculator question such as "write 2⁵ = 32 in log form" has the answer log₂(32) = 5. For a log calculator with different base, use change of base: log_b(x) = log(x) ÷ log(b).

Log calculator equation help: solve for x and find x

To use the log calculator to solve for x, rewrite the equation in exponent form:

log₃(x) = 4 → x = 3⁴ = 81. Use the antilog mode above (base 3, y = 4).
log(x) = 2.5 → x = 10^2.5 = 316.23.
2^x = 20 → x = log(20) ÷ log(2) = 4.3219. Here the log calculator finds x as a log with base 2.
log(x) + log(x − 3) = 1 → log(x(x − 3)) = 1 → x² − 3x − 10 = 0 → x = 5 (x = −2 is rejected because log of a negative number is undefined).

In a log x calculator, x must be greater than 0 and the base must be positive and not 1.

Log calculator trick CA Foundation students use (simple calculator)

CA Foundation students may only use a simple calculator with no log key. The log in calculator trick uses repeated square roots:

Enter the number, for example 2.
Press √ 15 times.
Subtract 1.
Multiply by 32,768 to get ln x, or by about 14,231 to get log₁₀(x).

For 2 this gives about 0.30103, which matches log 2. The trick works because x^1/32768 ≈ 1 + ln(x) ÷ 32,768, and 32,768 = 2¹⁵.

Log calculator for pH

pH = −log₁₀[H⁺]. A solution with [H⁺] = 3.2 × 10⁻⁵ mol/L has pH = −log(3.2 × 10⁻⁵) = 5 − 0.505 = 4.49. To go back, [H⁺] = 10^−pH, using the antilog mode.

Log P and log Kow calculator

In chemistry, log P (also written log K_ow) is the base-10 log of a compound's octanol–water partition coefficient. A partition coefficient of 1,000 gives log P = 3; a log Kow calculator is just log₁₀ of that ratio.

Log interpolation calculator

For values that change on a log scale, interpolate the logs, not the numbers. To find y at x = 30 between (10, 2) and (100, 4) on a log x scale: fraction = (log 30 − log 10) ÷ (log 100 − log 10) = 0.4771, so y = 2 + 0.4771 × 2 = 2.954.

Log likelihood calculator

In statistics, the log likelihood is the sum of the natural logs of the probabilities your model gives to the observed data. For three outcomes with probabilities 0.5, 0.8 and 0.9: ln 0.5 + ln 0.8 + ln 0.9 = −0.693 − 0.223 − 0.105 = −1.022. Use ln (base e) in the log calculator for each term.

Log calculator graph

The graph of y = log(x) passes through (1, 0) and (10, 1) for base 10, rises slowly, and is not defined for x ≤ 0. It is the mirror image of y = 10^x in the line y = x.

How does log work on a calculator, app or iPhone?

On a scientific calculator, "log" is base 10 and "ln" is base e; on most Casio models you press log, then the number. Some models have a log_▫▫ key for any base. On an iPhone, turn the Calculator app to landscape or open the scientific layout to find log₁₀ and ln. You do not need an app; this page works in the browser.

Log calculation in Java

For a log calculation in Java, Math.log10(x) gives base 10 and Math.log(x) gives the natural log. For any base use Math.log(x) / Math.log(b).

Log calculation questions

log₂(64) = 6
log(0.001) = −3
ln(e⁵) = 5
log 2 + log 5 = log 10 = 1
If log₅(x) = 3, x = 125

Frequently asked questions

What does log mean on a calculator?

The log key calculates the common logarithm, base 10. It tells you the power of 10 that gives your number: log 100 = 2. The ln key is the natural logarithm, base e.

How do I calculate log base 2?

Divide the log of the number by the log of 2: log₂(x) = log(x) ÷ log(2). For example log₂(32) = 1.50515 ÷ 0.30103 = 5. Or choose base 2 in this calculator.

How do I calculate a log with a different base?

Use the change of base formula: log_b(x) = ln(x) ÷ ln(b). Select Custom base in the calculator and it shows these steps.

What is the difference between log and ln?

log is base 10 and ln is base e (about 2.71828). ln x = 2.302585 × log x.

Can you take the log of zero or a negative number?

No. Logarithms are defined only for positive numbers, because no power of a positive base gives zero or a negative value.

What is antilog?

Antilog is the reverse of log. The antilog of y in base 10 is 10^y, so the antilog of 3 is 1,000. Choose Antilog in the calculator to find it.

What is log 2 and log 10?

log₁₀(2) ≈ 0.30103 and log₁₀(10) = 1. In natural logs, ln 2 ≈ 0.693147 and ln 10 ≈ 2.302585.

Is this log calculator free?

Yes. The log calculator is free, runs in your browser and supports base 10, base e, base 2 and any custom base.

How do I find log without a log key?

Take the square root 15 times, subtract 1 and multiply by about 14,231 for log base 10, or by 32,768 for ln.

How do I solve log(x) = y for x?

Rewrite it as x = 10^y (or b^y for base b) and use the antilog mode.

Can the log calculator convert to exponential form?

Yes. log_b(x) = y in exponential form is b^y = x. Use the log calculator exponential form check with the antilog mode: base 2, y = 5 gives 32.

Do I need a log calculator download or app?

No. There is no log calculator download; the page runs in any browser, including on phones.

Can I use the log calculator for algebra homework?

Yes. As a log calculator for algebra, it checks each step: expand with the log rules above, then evaluate each term here.

Is there a log calculator with base 2 and base 10?

Yes. Pick base 2, base 10, e or a custom base. The log calculator with base 2 gives log₂(1024) = 10.

What is the log calculator formula?

log_b(x) = log(x) ÷ log(b). The log calculator formula works for any positive base except 1.

Last updated 2026-09-18