Exponential Form Calculator (logarithmic form ↔ exponential form)

Understanding the exponential form calculator — concept and principle

This calculator converts between logarithmic form log_b(x) = y and exponential form b^y = x, and finds whichever of the three numbers is unknown: the logarithm, the result or the base. It also checks the answer with natural logarithms. Use it for algebra homework and for solving simple exponential and logarithmic equations.

A logarithm answers "what power of the base gives this number?"

How it is calculated

log_b(x) = y means exactly b^y = x (with b > 0, b ≠ 1, x > 0)

Unknown logarithm: y = ln x ÷ ln b (change of base)

Unknown result: x = b^y

Unknown base: b = x^(1/y)

log without a base means base 10; ln means base e ≈ 2.718. Enter 10 or 2.718281828 for b to use them.

The principle

A logarithm answers the question "to what power must b be raised to get x?". So the two forms say the same thing in two ways: the base stays the base, the logarithm becomes the exponent, and the argument becomes the result. Converting is the main tool for solving equations: an exponential equation is solved by taking a logarithm, and a logarithmic equation by rewriting it in exponential form.

Worked example

Default inputs: find the logarithm y, base 2, result 8.

Logarithmic form: log₂(8) = ?.
Since 2 × 2 × 2 = 8, the answer is 3.
Exponential form: 2³ = 8; logarithmic form: log₂(8) = 3.
Natural-log check: ln 8 ÷ ln 2 = 2.0794 ÷ 0.6931 = 3.
Base 2, result 8, exponent 3.

Each point is the calculator re-run with only “Result x (the argument of the log)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Result x (the argument of the log)	Answer
2	1
4	2
6	2.584962501
8	3
10	3.321928095
12	3.584962501
16	4

How to read the result

The logarithm grows slowly: log₂ 1,024 is only 10. A logarithm between two whole numbers means x is between two powers of b; for example log₂ 10 ≈ 3.32 because 10 lies between 8 and 16. Negative logarithms mean x is between 0 and 1.

Common mistakes

Swapping the base and the result when converting.
Using a base of 1 or a negative base.
Taking the logarithm of 0 or a negative number.
Reading log as base e; on most calculators log is base 10 and ln is base e.

Keep in mind: Results with irrational logarithms are rounded; keep exact forms such as log₂ 10 in written work.

Frequently asked questions

How do you convert a logarithm to exponential form?

log_b(x) = y becomes b^y = x.

What is log₂(8)?

3, because 2³ = 8.

How do I find a logarithm with an unusual base?

Use change of base: log_b(x) = ln x ÷ ln b.

Sources: OpenStax, College Algebra 2e (2021), §6.3 Logarithmic Functions

Checked 9 Nov 2026. OpenStax, College Algebra 2e (2021), §6.3 Logarithmic Functions — “Converting from Logarithmic to Exponential Form” and “Converting from Exponential to Logarithmic Form” (y = log_b(x) ⇔ b^y = x for x > 0, b > 0, b ≠ 1); §6.5 for the change-of-base formula.

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