Exponent, root and logarithm calculator
Raise numbers to any power, take nth roots, and compute logs in any base including ln and log₁₀, with the identities and steps shown.
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- Free, no account
Answer
Scientific notation
1.024000 × 10^3
Reciprocal (bⁿ)⁻¹
0.0009765625
Square of the result
1048576
Step-by-step working
A positive whole exponent means repeated multiplication.
2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024Check with logarithms.
log₁₀(result) = 10 × log₁₀(2) = 3.01029996The exponent rule log(bⁿ) = n·log(b) turns a big multiplication into an easy addition.
Four related calculators
- Powers — raise any base to any exponent, including negative and fractional ones.
- Roots — square, cube and nth roots, with odd roots of negative numbers supported.
- Logarithms — any base, via the change-of-base rule, plus ln, log₁₀ and log₂ alongside.
- Scientific notation — convert in both directions, with engineering notation and significant figures.
Powers, roots and logs are three views of the same relationship. If bⁿ = x, then the nth root of x is b and log base b of x is n. Each tab answers for a different unknown.
Exponent rules
| Rule | Statement | Example |
|---|---|---|
| Product | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 2⁷ ÷ 2⁴ = 2³ = 8 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (2³)² = 2⁶ = 64 |
| Zero exponent | b⁰ = 1 for b ≠ 0 | 7⁰ = 1 |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent | b^(m/n) = ⁿ√(bᵐ) | 8^(2/3) = (∛8)² = 4 |
The negative and fractional rules are not conventions invented for convenience — they are forced by the product rule. If b³ × b⁻³ has to equal b⁰ = 1, then b⁻³ can only be the reciprocal of b³.
Logarithm rules
| Rule | Statement | Example |
|---|---|---|
| Product | log(xy) = log x + log y | log₁₀(2 × 50) = 0.301 + 1.699 = 2 |
| Quotient | log(x ÷ y) = log x − log y | log₂(32 ÷ 4) = 5 − 2 = 3 |
| Power | log(xⁿ) = n · log x | log₁₀(10³) = 3 × 1 = 3 |
| Change of base | log_b(x) = ln x ÷ ln b | log₂(1024) = 6.931 ÷ 0.693 = 10 |
| Identity | log_b(b) = 1 and log_b(1) = 0 | log₇(7) = 1 |
Turning multiplication into addition is precisely why logarithms were invented in the early 1600s, and why slide rules worked. The change-of-base rule is what lets this calculator handle a base of 3, 7 or 1.05 without needing a separate function for each.
Why some inputs are rejected
- log of zero or a negative number — no power of a positive base ever produces zero or a negative result, so there is no real answer. As x approaches zero from above, log x heads to −∞.
- Base 1 logarithms — 1 to any power is 1, so the question “what power of 1 gives 8?” has no answer.
- Even roots of negatives — any real number squared is non-negative, so √(−4) is not real. Odd roots are fine: ∛(−27) = −3.
- Zero to a negative power — that is 1 ÷ 0, which is undefined.
- Negative base with a fractional exponent — (−8)^0.5 would be an even root of a negative number.
Scientific notation
Scientific notation writes a number as a coefficient between 1 and 10, multiplied by a power of ten. Move the decimal point until one non-zero digit sits in front of it, and count the moves: left gives a positive exponent, right gives a negative one.
| Number | Scientific | Engineering |
|---|---|---|
| 149,600,000 | 1.496 × 10⁸ | 149.6 × 10⁶ |
| 0.000 000 45 | 4.5 × 10⁻⁷ | 450 × 10⁻⁹ |
| 602,214,076,000,000,000,000,000 | 6.02214076 × 10²³ | 602.214076 × 10²¹ |
| 0.0025 | 2.5 × 10⁻³ | 2.5 × 10⁻³ |
Engineering notation restricts the exponent to multiples of three so it lines up with the SI prefixes — kilo, mega, giga, milli, micro, nano — which is why electronics and mechanical work prefer it.
Where logs show up in real units
- Decibels — 10·log₁₀ of a power ratio. Ten decibels is a tenfold change in power.
- pH — the negative base-10 log of hydrogen ion concentration, so pH 4 is ten times more acidic than pH 5.
- Richter magnitude — base 10, so a magnitude 6 releases about 32 times the energy of a magnitude 5.
- Algorithmic complexity — binary search takes log₂(n) steps, which is why doubling the data adds only one step.
- Compound growth — the time to reach a target is ln(target ÷ start) ÷ ln(1 + rate).
Precision
Calculations use double-precision floating point, about 15 significant digits, with results displayed to ten decimal places and trailing zeros trimmed. Very large or very small results switch automatically to exponential form. Powers larger than roughly 1.8 × 10³⁰⁸ overflow what a double can hold, and the tool says so rather than showing “Infinity”. Everything runs in your browser; nothing is uploaded.
Questions
How do I take a log in an unusual base?
Enter the base and the number and the calculator applies the change-of-base rule, log_b(x) = ln(x) ÷ ln(b), showing both natural logs so you can verify the division yourself.
Can it handle negative and fractional exponents?
Yes. A negative exponent becomes a reciprocal and a fractional one becomes a root, and both rewrites are shown as steps before the numeric answer.
Why does an even root of a negative number fail?
No real number squared gives a negative result, so the even root is undefined over the reals. Odd roots of negatives are fine and the tool computes them, for example the cube root of −27 is −3.
What does the scientific notation tab do?
It converts between plain decimal and a × 10ⁿ form in both directions, reports the exponent and the number of significant figures, and gives the engineering-notation variant.