Logarithm Calculator

1. Enter the number x, which must be positive.
2. Enter the base b. Use 10 for common log, e for natural log, or any other base.
3. Read the logarithm, the power the base is raised to.
4. See the natural log and log base 10 alongside.

Here is what each result means:

A logarithm answers the question: to what power must the base be raised to get this number? In symbols, log_b(x) = y means b^y = x. So log_2(8) = 3, because 2^3 = 8. A logarithm is the inverse of an exponent.

Logarithms turn multiplication into addition and huge ranges into manageable scales. That is why they power the decibel scale for sound, the Richter scale for earthquakes, and pH in chemistry. The two most common are the common log, base 10, and the natural log, base e.

It uses the change-of-base formula, which turns a log in any base into a ratio of natural logs.

1. Take the natural log of the number and of the base.
2. Divide the first by the second to get the log in your base.
3. Report the natural log and log base 10 as common references.

Because a log undoes a power, the exponent calculator is its natural companion.

Find log_2(8) and log_10(1000).

Calculation: for the first, ask what power of 2 gives 8. Since 2^3 = 8, the answer is 3. For the second, 10^3 = 1000, so log_10(1000) = 3 as well. Both work out to 3 because each base cubed reaches its number.

Two bases appear far more than any others, and each has its own notation.

When a base is not shown, log usually means base 10 in general use and base e in higher maths. This tool shows both so there is no ambiguity.

A few identities make logs easier to work with by hand.

These turn multiplication and powers into addition, which is the whole reason logarithms were invented.

A Larger Base Gives a Smaller Log

Reaching the same number takes a smaller power when the base is bigger, so the logarithm falls.

The Number Relative to the Base

If the number equals the base the log is 1; if it equals 1 the log is 0, whatever the base.

Numbers Below One

The log of a number between 0 and 1 is negative, since the base must be raised to a negative power.

Homework and Algebra

Evaluate logs in any base, including bases your calculator lacks a button for.

Science and Scales

Work with pH, decibels or the Richter scale, which are all logarithmic.

Growth and Finance

Use natural logs to solve for time in exponential growth or compound interest.

1. Taking the Log of Zero or a Negative

These are undefined for real numbers. The input must be greater than zero.

2. Confusing Log and Ln

log usually means base 10 and ln means base e. Mixing them changes the answer.

3. Using Base 1

A base of 1 has no valid logarithm, since 1 to any power is always 1.

4. Forgetting Negative Results

Numbers below one give negative logs. That is correct, not an error.

5. Misapplying the Rules

log(a + b) is not log a plus log b. The product rule applies to multiplication, not addition.

The change-of-base formula is exact; only the displayed decimals are rounded.

A logarithm is the power a base must be raised to in order to reach a number. So the log base 2 of 8 is 3, because 2 to the 3rd is 8. It is the inverse of an exponent.

Use the change-of-base formula: divide the natural log of the number by the natural log of the base. This tool does that, so you can find a log in any base your device lacks a button for.

log usually means the common logarithm, base 10, while ln means the natural logarithm, base e, about 2.718. They give different values for the same number, so keep track of the base.

Not for a real result. The logarithm of zero or a negative number is undefined over the real numbers, so the input must be greater than zero.

The logarithm of 1 is 0 in every base, because any base raised to the power zero equals 1. Likewise, the log of the base itself is always 1.

When the number is between 0 and 1, the base must be raised to a negative power to reach it, so the logarithm is negative. That is a correct result, not an error.

The natural log, base e, appears in growth and decay, compound interest and calculus. It is the natural choice whenever a quantity changes at a rate proportional to its size.

The product rule turns log of a product into a sum of logs, the quotient rule into a difference, and the power rule brings an exponent to the front. They turn multiplication and powers into addition.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Logarithm (Wikipedia).
• Logarithms explained (Maths Is Fun).
• The natural logarithm (Maths Is Fun).