Exponent and logarithm calculator guide
An exponent tells you how many times a base is used as a factor: 2^10 multiplies 2 by itself ten times to give 1024. Exponents extend beyond whole numbers — a negative exponent produces a reciprocal (2^-3 = 1/8), and a fractional exponent produces a root (9^0.5 = 3).
A logarithm answers the inverse question: to what power must the base be raised to produce a given number? Because 2^10 = 1024, log_2(1024) = 10. This calculator evaluates both operations and accepts any valid base, so it handles common logarithms (base 10), natural logarithms (base e ≈ 2.71828), binary logarithms (base 2), and everything in between via the change-of-base rule.
How to use this exponent and logarithm calculator
- Choose a mode: Select Power (x^y) to evaluate an exponent, or Logarithm (log_b x) to find the exponent that produces a number.
- Enter the values: For a power, enter the base and the exponent. For a logarithm, enter the base (positive, not 1) and the argument (positive).
- Calculate: Press Calculate. The result panel shows the evaluated expression and its value to eight significant digits.
- Check your work: Verify a logarithm by raising the base to the result in Power mode — you should get the original argument back.
Formula and variables
For whole-number exponents the power is repeated multiplication; the definition extends to negative exponents as reciprocals and to fractional exponents as roots. The logarithm is computed with the change-of-base rule: divide the natural logarithm of the argument by the natural logarithm of the base. The base must be positive and not equal to 1, and the argument must be positive.
x^y = x × x × … × x (y factors); log_b(x) = ln(x) ÷ ln(b) ⟺ b^(log_b(x)) = x- x — Base (power mode)
- The number being raised to a power. (unitless)
- y — Exponent
- The power applied to the base; may be negative or fractional. (unitless)
- b — Logarithm base
- Must be positive and not equal to 1. Use 10 for common log, e ≈ 2.718281828 for natural log, 2 for binary log. (unitless)
- ln — Natural logarithm
- The logarithm to base e, used internally for the change-of-base rule. (unitless)
Worked example: solving for an unknown exponent
An investment doubles repeatedly. How many doubling periods are needed for $1,000 to grow to $16,000? This asks for log_2(16), because 16,000 ÷ 1,000 = 16.
- Mode
- Logarithm
- Base (b)
- 2
- Argument (x)
- 16
- log_2(16) = ln(16) ÷ ln(2)
- ln(16) ≈ 2.7725887 and ln(2) ≈ 0.6931472
- 2.7725887 ÷ 0.6931472 = 4
- Check in Power mode: 2^4 = 16
Result: log_2(16) = 4
Four doubling periods are required. Logarithms convert questions about repeated multiplication into single divisions, which is why they appear throughout compound growth, pH, decibels, and information theory.
Understanding your results
Power results
The result is the base multiplied by itself the indicated number of times. A negative exponent means a reciprocal: 2^-3 = 1/8 = 0.125. A fractional exponent means a root: 27^(1/3) = 3. Any nonzero base to the power 0 equals 1.
Logarithm results
The result is the exponent to which the base must be raised to produce the argument. A result greater than 1 means the argument exceeds the base; a result between 0 and 1 means the argument lies between 1 and the base; a negative result means the argument is a fraction below 1.
Significant digits
Results are displayed to eight significant digits. For irrational results such as log_2(10) ≈ 3.3219281, the displayed value is rounded; use the exact expression, not the rounded value, in further work.
Assumptions
- Inputs and outputs are real numbers; complex results are out of scope.
- Logarithms are computed with the change-of-base rule log_b(x) = ln(x) ÷ ln(b), matching the NIST Digital Library of Mathematical Functions definition (DLMF 4.2.8).
- The logarithm base is positive and not equal to 1, and the argument is positive.
- Powers follow IEEE 754 double-precision arithmetic, so very large or very small magnitudes lose precision or overflow.
Limitations
- A negative base with a fractional exponent (for example (-8)^(1/3)) has no real power defined by this calculator, even though a real cube root exists; rewrite it as -(8^(1/3)).
- The expression 0^0 is returned as 1, following the standard computing convention, although the expression is indeterminate in continuous mathematics.
- Results beyond roughly 1.8 × 10^308 overflow and are rejected rather than shown as Infinity.
- The calculator evaluates single expressions only; it does not solve equations, expand logarithmic identities, or show algebraic steps.
Common mistakes
- Using base 1 for a logarithm. The function 1^y is constant, so log base 1 is undefined; the base must be positive and different from 1.
- Taking the logarithm of zero or a negative number. The domain of a real logarithm is positive numbers only.
- Confusing log base 10 (common log) with ln (base e ≈ 2.71828) — enter the base explicitly instead of assuming.
- Misreading x^-y as a negative result; a negative exponent gives a positive reciprocal, e.g. 5^-2 = 0.04.
- Believing (a + b)^n equals a^n + b^n; exponents do not distribute over addition.
- Entering the exponent and base in the wrong fields: 2^10 = 1024 but 10^2 = 100.
Practical use cases
Growth and decay problems
Evaluate compound growth factors such as 1.05^20 for a 5% annual rate over 20 years, or find the time in log_2(16) = 4 doubling periods for a quantity to grow sixteenfold.
Science and engineering scales
Logarithmic scales such as pH, decibels, and earthquake magnitude are base-10 logarithms; computing log_10 of a ratio converts a multiplicative difference into an additive one.
Computer science
Binary logarithms (base 2) describe algorithm complexity and information content; log_2(1024) = 10 bits are needed to represent 1024 distinct values.
Planning and decision guide
Exponent and logarithm are inverse operations
Because b^y = x means exactly the same thing as log_b(x) = y, each mode can verify the other. If log_3(81) = 4, then 3^4 must equal 81 — a quick way to check either answer.
Choose the base that matches the problem
Use base 10 for decimal scales like pH and decibels, base 2 for doubling problems and binary information, and base e ≈ 2.718281828 for continuous growth and calculus. The change-of-base rule means any base can be converted to any other, so a calculator with a free base input replaces three separate log keys.
Fractional and negative exponents follow the same rules
x^(1/n) is the nth root of x, and x^(-n) is 1 ÷ x^n. These two extensions let a single power mode compute square roots (exponent 0.5), cube roots (exponent 1/3), reciprocals (exponent -1), and combinations such as 8^(-2/3) = 1/4.
Frequently asked questions
What is an exponent?
An exponent counts how many times a base is multiplied by itself. In 2^10, the base 2 is multiplied ten times, giving 1024. Negative exponents give reciprocals and fractional exponents give roots.
What is a logarithm?
A logarithm is the exponent to which a base must be raised to produce a given number. Because 10^2 = 100, log base 10 of 100 is 2. It is the inverse operation of exponentiation.
Why can the logarithm base not be 1 or negative?
The powers of 1 are always 1, so no exponent can produce any other value and log base 1 is undefined. A negative base does not produce a consistent real-valued exponential function. Valid bases are positive numbers other than 1.
How do I calculate a logarithm to a base my calculator does not have?
Use the change-of-base rule: log_b(x) = ln(x) ÷ ln(b), or equivalently log_10(x) ÷ log_10(b). For example log_2(10) = ln(10) ÷ ln(2) ≈ 3.3219281.
What is the difference between log and ln?
log without a base usually means the common logarithm, base 10, while ln is the natural logarithm, base e ≈ 2.71828. Enter 10 or 2.718281828 as the base to match each convention.
What does a negative exponent mean?
A negative exponent produces the reciprocal of the positive power: x^-n = 1 ÷ x^n. For example, 2^-3 = 1/8 = 0.125.
Can I compute roots with this calculator?
Yes. A fractional exponent is a root: the square root of 9 is 9^0.5 = 3, and the cube root of 27 is 27^(1/3) ≈ 27^0.3333333 ≈ 3.
Why does the calculator reject (-8)^(1/3) even though the answer is -2?
A negative base with a fractional exponent is generally undefined in the real-number power function, which is defined through logarithms of positive numbers. Rewrite the expression as -(8^(1/3)) to get -2.
Sources and review
- DLMF §4.2 Logarithm, Exponential, Powers — NIST Digital Library of Mathematical Functions. Accessed 2026-08-19.
- Algebra and Trigonometry 2e, Section 6.3: Logarithmic Functions — OpenStax, Rice University. Accessed 2026-08-19.
Reviewed 2026-08-19.