Exponent, Root and Logarithm Calculator

What you can work out here

This calculator finds a power $x^y$, an nth root $\sqrt[n]{x}$, a logarithm $\log_b x$ in any base, or the exponent $y$ that solves $b^y = x$. The four are inverses of one another: $2^5 = 32$, $\sqrt[5]{32} = 2$ and $\log_2 32 = 5$ describe the same fact from three sides. Answers are exact whenever an exact answer exists and fits on the page (whole numbers with every digit, fractions, simplest radical forms such as $6\sqrt{2}$, and logarithms such as $\log_4 8 = 3/2$). Otherwise they are decimals rounded to the number of significant figures you choose. When there is no real answer, the result says so and explains why instead of showing an error.

How to use it

Choose what to Calculate (Power, Root, Log or Solve bʸ = x) and type the numbers. The answer updates as you type.

• Base and Exponent (Power): each box takes a whole number, a decimal, a fraction such as 2/3 or -1/2, a mixed number such as 2 1/2, or scientific notation such as 1.2e-5. The base can also be e, for powers of Euler’s number. Write a negative base as -8 or (-8).
• Which root and Number (Root): square and cube roots have their own buttons; Other index asks for $n$.
• Log base and Number (Log): base 10, base $e$ and base 2 are buttons; Other base opens the Base box.
• Base and Number (Solve bʸ = x): $b$ and $x$.
• Answer format: significant figures (1 to 30; 10 when left blank) and automatic or scientific notation.

The Try chips load cases from this page. Under each answer are the steps with your numbers, a table of neighboring powers or roots with a CSV download, and the curve through your answer. Switching modes keeps your numbers, so after $\log_2 1{,}000{,}000$ you can press Solve and see the same answer as the exponent in $2^y = 1{,}000{,}000$.

How to calculate exponents, including negative and fractional powers

A whole-number exponent counts repeated multiplication, a negative exponent takes the reciprocal, and a fractional exponent $p/q$ takes the $q$th root and then the $p$th power:

$$x^n = \underbrace{x \times x \times \cdots \times x}_{n}, \qquad x^{-n} = \frac{1}{x^n}, \qquad x^{p/q} = \left(\sqrt[q]{x}\right)^p = \sqrt[q]{x^p}, \qquad x^0 = 1$$

• $x$ is the base and $n$ a whole number;
• $p/q$ is the exponent as a fraction in lowest terms.

So $2^5 = 32$, $5^{-3} = 1/5^3 = 1/125 = 0.008$, and $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$. The calculator reads an exponent typed as a decimal as the exact fraction it stands for: 0.75 is 3/4, 0.2 is 1/5, and 0.3333 is 3,333/10,000, not 1/3. For positive bases that only matters in the last digits, but for negative bases it decides whether there is an answer at all (see below).

Worked example: 12 to the power 3/2

Enter Base 12 and Exponent 3/2.

1. The exponent 3/2 means the square root, then the cube: $12^{3/2} = (\sqrt{12})^3$.
2. Write it as one root: $12^{3/2} = \sqrt{12^3} = \sqrt{1{,}728}$.
3. Pull out the largest perfect square: $1{,}728 = 24^2 \times 3$, so $\sqrt{1{,}728} = 24\sqrt{3}$.
4. As a decimal to 10 significant figures: $24\sqrt{3} \approx 41.56921938$.

The table shows the neighboring powers of 12 at 1/2, 5/2 and so on, each 12 times the one before. Switch modes and the same numbers answer the inverse questions: Root gives $\sqrt{1{,}728} = 24\sqrt{3}$ again, and Log and Solve keep the base 12 and give $\log_{12} 1{,}728 = 3$, the exponent in $12^3 = 1{,}728$.

How to find a square root, cube root or nth root

The nth root of $x$ is the number that gives $x$ when raised to the power $n$; it equals $x^{1/n}$. The root sign always means the principal root. For an even index that is the root that is 0 or more: $\sqrt{9} = 3$, even though $(-3)^2 = 9$ as well. For an odd index the root has the same sign as the number, so $\sqrt[3]{-27} = -3$.

A perfect power has an exact root, which the calculator gives as a whole number or fraction. Anything else has an irrational root, shown in simplest radical form and as a rounded decimal.

Simplifying roots: simplest radical form

To simplify $\sqrt[n]{m}$, split $m$ into a perfect nth power times a leftover with no nth-power factor, then take the root of the perfect power:

• $72 = 6^2 \times 2$, so $\sqrt{72} = 6\sqrt{2} \approx 8.485281374$.
• $54 = 3^3 \times 2$, so $\sqrt[3]{54} = 3\sqrt[3]{2} \approx 3.779763150$.
• $\sqrt{1/2} = \sqrt{2}/\sqrt{4} = \sqrt{2}/2$ (about 0.7071067812), which clears the root from the denominator.

The steps show the split the calculator used, so you can check it against your own factoring.

How to calculate a logarithm in any base (change of base)

$\log_b x$ is the exponent that turns $b$ into $x$: $\log_b x = y$ means $b^y = x$. The base must be greater than 0 and not 1, and $x$ must be greater than 0. For any base, divide natural logs:

$$\log_b x = \frac{\ln x}{\ln b}$$

For $\log_2 1{,}000{,}000$:

$$\log_2 1{,}000{,}000 = \frac{\ln 1{,}000{,}000}{\ln 2} = \frac{13.81551056}{0.6931471806} \approx 19.93156857$$

That fits between whole powers: $2^{19} = 524{,}288$ and $2^{20} = 1{,}048{,}576$.

Some logarithms are exact fractions, and the calculator finds them. $\log_4 8 = 3/2$, because $8 = 2^3$ and $4 = 2^2$, so $8 = 4^{3/2}$. Likewise $\log_{10} 0.001 = -3$.

Natural log (ln) vs common log (log)

They are the same idea in different bases. The natural log, ln, uses base $e \approx 2.718281828$; the common log, usually written log, uses base 10. So $\ln 20 \approx 2.995732274$ while $\log_{10} 20 \approx 1.301029996$, and the two always differ by the same factor, $\ln 10$. In some computer science writing a bare “log” means base 2 instead, so check which base a formula assumes. In a spreadsheet, =LN(x) is the natural log and =LOG(x, base) works in any base, using base 10 when you leave the base out.

An antilog undoes a log: the antilog of 2.5 in base 10 is $10^{2.5} \approx 316.2277660$, and the natural antilog of 1.5 is $e^{1.5} \approx 4.481689070$. Use Power with base 10 or e.

Solving for an exponent: bʸ = x

Taking logs of both sides gives $y = \log_b x = \ln x \div \ln b$. This answers “how many times do I multiply by $b$ to reach $x$?”

• Doublings to reach 1,000,000: $2^y = 1{,}000{,}000$ gives $y \approx 19.93156857$, so 20 whole doublings are needed ($2^{20} = 1{,}048{,}576$).
• Growth of 7% per step: $1.07^y = 2$ gives $y \approx 10.24476835$. After 10 steps the total is $1.07^{10} \approx 1.967151357$, short of double; the 11th step reaches $1.07^{11} \approx 2.104851952$.

When the answer is between 0 and 24, the table in Solve mode lists every whole power from $b^0$ up to one past the answer, so you can see the step where $x$ is passed; otherwise it shows the eight whole powers around the answer.

When there is no real answer (negative roots, log of zero)

• Even roots of negative numbers. No real number squared is negative, so $\sqrt{-9}$ has no real value. The calculator says so and gives the complex value for information: $\sqrt{-9} = 3i$, where $i = \sqrt{-1}$.
• Negative bases with fractional exponents. A real answer exists only when the exponent in lowest terms has an odd denominator: $(-8)^{1/3} = -2$ and $(-8)^{0.2} = -\sqrt[5]{8} \approx -1.515716567$, because 0.2 is 1/5. But 0.3333 is 3,333/10,000, whose denominator is even, so $(-8)^{0.3333}$ has no real value (its principal complex value is about $1.000112050 + 1.731826033i$). If you meant the cube root, type 1/3.
• Logarithms of 0 and of negative numbers. No power of a positive base is 0 or negative, so $\log_b 0$ has no value at all and $\log_2(-8)$ has no real value (its principal complex value is $3 + 4.532360142i$).
• Base 1, or a base of 0 or less, in Log and Solve, and 0 to a negative power, which would mean dividing by 0: these can’t be calculated, and the message under the box says why.

What is 0 to the power of 0?

This calculator gives $0^0 = 1$ and adds a note. That value keeps the binomial theorem working, which is Donald Knuth’s argument for it in the paper listed in the sources. Some algebra courses, including the OpenStax text in the sources, leave $0^0$ without a value, and in calculus it is an indeterminate form: when a base and an exponent both only approach 0, the limit can come out as any number. So if your course leaves $0^0$ without a value, follow your course.

Very large and very small results

Whole-number results are exact up to about 30,000 digits, listed in full with a button that copies them. For $2^{100}$ that is all 31 digits:

$$2^{100} = 1{,}267{,}650{,}600{,}228{,}229{,}401{,}496{,}703{,}205{,}376$$

Above about $1.8 \times 10^{308}$, ordinary double-precision arithmetic overflows to Infinity. When a result that large is too long to list, or isn’t a whole number or fraction at all (such as $7^{1{,}000.5}$), the calculator works with its logarithm instead. For $2^{1{,}000{,}000}$:

$$\log_{10} 2^{1{,}000{,}000} = 1{,}000{,}000 \times \log_{10} 2 \approx 301{,}029.9957$$

The whole part of the logarithm gives the power of ten, and 10 raised to the rest gives the leading digits, so the result is about $9.900656229 \times 10^{301{,}029}$.

Reading the result

• The answer is exact when it says so; otherwise it is rounded to your significant figures. Irrational roots also show their simplest radical form.
• The tiles add the exact form, scientific notation, the negative square root, the digit count, or the same number in other log bases, depending on the mode.
• The steps show the method with your numbers and end with the spreadsheet formula that does the same calculation.
• The table gives nearby powers (Power), roots of your number from square to 10th (Root), or whole powers of the base around your number (Log and Solve).
• The check puts the rounded answer back in, so it can differ from your number in the last few digits, more so when the answer is a large number.

Assumptions and limitations

• Only real numbers count as answers. Complex values appear for information only.
• Decimals are rounded half up from a calculation carried well past the 30 figures the page can show, so the last figure shown is correctly rounded; exact forms are never rounded.
• Exponents are limited to ±1,000,000,000 and root indexes to 1,000. Simplest radical form is found for numbers under the root up to about 9 × 10¹⁵; above that the calculator gives the decimal only.
• The calculator works one operation at a time. For expressions such as $3 \times 2^4 - 1$, use a scientific calculator.

Common mistakes

• Reading $-3^2$ as $(-3)^2$. Order of operations applies the exponent first, so $-3^2 = -(3^2) = -9$, while $(-3)^2 = 9$. Type a negative base as -3 (or (-3)) to mean $(-3)^2$.
• Typing 0.333 for 1/3. With a positive base the difference is small, but with a negative base it removes the real answer; the Exponent box warns you when it spots this.
• Forgetting the negative square root. $\sqrt{72}$ is the positive root, 8.485281374; when solving $x^2 = 72$, $x = -8.485281374$ works too.
• Mixing up log and ln. A result that is off by a factor of about 2.302585093 means one of the two used base $e$ and the other base 10.
• Dividing instead of taking a root. $64^{1/2}$ is $\sqrt{64} = 8$, not $64 \div 2 = 32$.