Logarithm Calculator Guide

What This Calculator Does

The Logarithm Calculator answers two inverse questions:

• Given a base b and value x, find the exponent log_b(x).
• Given a base b and exponent y, evaluate the power b^y.

The page has a Logarithm mode and a Power mode. Logarithm mode shows the custom-base result, reference values for ln(x) and log10(x), and the relationship b^y=x so you can check the setup. It is useful for classwork, homework checks, change-of-base calculations, and basic exponential-growth verification.

If you need parentheses, functions, or several operators in one formula, use the Mathematical Expression Calculator. For a keypad workflow with trig, ln, log, and exponent keys, use the Scientific Calculator. For reporting a result to a chosen number of decimal places, use the Rounding Calculator.

When to Use It

• You need log_2(32), log_10(0.01), or log_3(10).
• You want a valid custom base rather than only base 10 or base e.
• You want to compute a power and then verify the exponent with a logarithm.
• You want to compare a custom-base logarithm with the natural and common logarithms of the same value.

This is not an equation solver, logarithm simplifier, or complex-number calculator. It expects one base and one value or exponent at a time.

Inputs Explained

Mode

• Find Logarithm: enter a base and value to calculate log_b(x).
• Evaluate Power: enter a base and exponent to calculate b^y.

When you switch modes, the second input changes from Value to Exponent. Once the required fields contain valid numbers, the page updates the result; Calculate lets you confirm it again.

Base

The base is b in the logarithm definition. In Logarithm mode it must satisfy b>0 and b≠1. A base between 0 and 1 is valid, such as 0.5, but the logarithm is decreasing, so check the sign through the relationship b^y=x.

Value or Exponent

The Logarithm-mode value x must be greater than 0. In Power mode, the exponent can be negative or fractional as long as b^y produces a finite real result.

The input format is strict ordinary decimal notation: 1000, -2, 3.5, .5, and 3. work. 1e3, comma-separated numbers, variables, units, function calls, and expressions such as 2+3 do not.

How the Calculation Works

Logarithm mode: change of base

The page evaluates any valid base with the change-of-base identity:

log_b(x) = ln(x) / ln(b)

For example:

log_2(32) = ln(32) / ln(2) = 5

The displayed ln(x) and log10(x) values are references for the same value; they do not replace the custom base in the main result.

Power mode: the inverse relationship

Power mode directly evaluates:

b^y

When the base is positive, not 1, and the result is positive, the result card also shows the reverse check log_b(b^y)=y. A negative base with a fractional exponent may not produce a real number, so the page reports an error rather than interpreting complex values.

Example

Example 1: an integer logarithm

In Logarithm mode, enter base 2 and value 32:

log_2(32) = 5

The verification line is 2^5=32. The result card also lists ln(32) and log10(32) as reference values.

Example 2: a value below 1

Enter base 10 and value 0.01:

log_10(0.01) = -2

The negative answer is correct because 10^-2=0.01.

Example 3: a non-integer custom-base result

With base 3 and value 10, the page displays about 2.0959032743. That decimal is expected: there is no integer exponent whose power of 3 is exactly 10, but the result still satisfies the change-of-base calculation.

Example 4: verify with power mode

Switch to Power mode and enter base 2 and exponent 5; the result is 32. Return to Logarithm mode with base 2 and value 32 to recover 5 and verify the inverse relationship.

How to Understand the Result

Main result

In Logarithm mode the main result is the exponent y; in Power mode it is the value of b^y. Do not read a logarithm result as the original value.

ln and log10 references

ln(x) uses base e, while log10(x) uses base 10. They provide familiar comparison points when the main result uses a custom base.

Verification relationship

The result card makes the check explicit: Logarithm mode shows log_b(x)=y and b^y=x; Power mode shows b^y=result and, when applicable, log_b(result)=y.

Display precision

Ordinary finite results are displayed to roughly 10 decimal places. Very small or very large magnitudes switch to six-digit scientific notation. The page uses JavaScript Number arithmetic, not arbitrary precision, so long decimals are approximations.

Common Mistakes

• Using 1 as a logarithm base; it cannot uniquely determine an exponent.
• Entering 0 or a negative value in Logarithm mode.
• Assuming log always means base 10 when the page's main base is the value you entered; log10(x) is the separate reference card.
• Treating log_b(x) as if it returned x instead of checking b^y=x.
• Entering 1e3 or 2+3 even though the page expects ordinary decimal numbers.
• Assuming every negative-base and fractional-exponent combination is a real power.

FAQ

What is a logarithm?

log_b(x) is the exponent y satisfying b^y=x. For example, log_2(8)=3 because 2^3=8.

Why cannot the base be 1?

Every real power of 1 is still 1, so it cannot uniquely recover an exponent. That is why log_1(x) is not a valid real logarithm.

What is the difference between log, ln, and log10?

The main result uses the custom base you enter. ln(x) uses base e, and log10(x) uses base 10. In textbooks, an unqualified log often means base 10, but follow the convention stated by your problem.

Why is the answer a decimal?

Only powers that land exactly on the value produce an integer logarithm. log_3(10) naturally corresponds to a non-integer exponent, so a decimal approximation is expected.

Can this solve log_2(x)=5?

It can verify the result by showing 2^5=32, but it does not parse equation text or solve for an unknown. Use an equation solver or rearrange the equation first.

Notes

Logarithm mode requires b>0, b≠1, and x>0. Power mode uses JavaScript Math.pow, and the result must be finite; negative bases, negative exponents, fractional exponents, and very large values need extra checking.

Valid inputs and results are synchronized to URL parameters, so copying the link can restore the same input set. That is not calculation history and does not store a list of prior results. The page does not support variables, expressions, units, text scientific notation, symbolic log laws, equation solving, graphs, or complex-number explanations. For engineering, finance, chemistry, or research work, confirm the base, units, and rounding rule required by the domain.