Exponential Form Calculator
Choose a mode and enter your values. The calculator rewrites any whole number as a product of prime powers, converts any number to scientific notation, rewrites a logarithm as an equivalent exponential equation, or converts an exponential equation back to logarithm form. Every result includes a step-by-step worked solution.
Formula
Worked example
Prime mode: 360 = 2 x 2 x 2 x 3 x 3 x 5 = 2^3 x 3^2 x 5. Scientific mode: 45000 = 4.5 x 10^4. Log mode: log_2(8) = 3 converts to 2^3 = 8. Exponential mode: 2^5 = 32 converts to log_2(32) = 5.
What is exponential form?
Exponential form is a way of writing a number using a base and an exponent rather than spelling out every factor. For whole numbers this means prime factorization: for example, 360 = 2^3 x 3^2 x 5, which shows at a glance that 360 is divisible by 2 three times, by 3 twice, and by 5 once. For very large or very small numbers, exponential form means scientific notation: any real number is written as a coefficient between 1 and 10 multiplied by a power of ten, for example 45,000 = 4.5 x 10^4 or 0.00045 = 4.5 x 10^-4. In algebra, exponential form refers to the equivalent of a logarithm equation: if log base b of c equals a, the exponential form is b^a = c.
How to convert a number to exponential form (prime factorization)
Start dividing the number by the smallest prime (2). Keep dividing until 2 no longer divides evenly, then move to the next prime (3, 5, 7, ...). Continue until the remaining quotient is 1. Count how many times each prime appears - that count is the exponent. Write the result as a product of base-exponent pairs. For 360: 360/2 = 180, 180/2 = 90, 90/2 = 45, 45/3 = 15, 15/3 = 5, 5/5 = 1. Tally: 2 appears 3 times, 3 appears 2 times, 5 appears once. Result: 2^3 x 3^2 x 5. This process is guaranteed to terminate and give a unique result by the Fundamental Theorem of Arithmetic.
Converting between logarithm and exponential form
Logarithms and exponential equations are two ways to state the same mathematical relationship. The logarithm form log_b(c) = a says: "b raised to the power a equals c". The exponential form says exactly the same thing: b^a = c. Converting between the two is therefore a simple relabeling of the same identity. This equivalence is the foundation of solving exponential equations in algebra: isolate the exponential expression, take the log of both sides, then convert to log form and solve. Common special cases: log base 10 is written "log", log base e (Euler's number 2.71828...) is written "ln" (the natural logarithm), and 2^10 = 1024 converts to log_2(1024) = 10.
Scientific notation and powers of ten
Scientific notation expresses any non-zero number as c x 10^n where 1 is at most the absolute value of c and that value is less than 10. The exponent n counts how many places the decimal point shifts: positive n means the original number is 10^n times larger than the coefficient (large number), negative n means it is 10^n times smaller (small number). For example, the speed of light is approximately 3 x 10^8 metres per second, and the mass of a proton is about 1.67 x 10^-27 kilograms. Multiplying two numbers in scientific notation is done by multiplying their coefficients and adding their exponents. Dividing uses the quotient of coefficients and the difference of exponents.
Common exponential form examples
| Number | Prime factorization | Scientific notation | Notes |
|---|---|---|---|
| 2 | 2^1 | 2 x 10^0 | Smallest prime |
| 8 | 2^3 | 8 x 10^0 | Perfect cube |
| 12 | 2^2 x 3 | 1.2 x 10^1 | |
| 36 | 2^2 x 3^2 | 3.6 x 10^1 | Perfect square |
| 100 | 2^2 x 5^2 | 1 x 10^2 | Powers of 10 |
| 360 | 2^3 x 3^2 x 5 | 3.6 x 10^2 | Highly composite |
| 1000 | 2^3 x 5^3 | 1 x 10^3 | 10^3 |
| 0.001 | - | 1 x 10^-3 | Small decimal |
| 0.00045 | - | 4.5 x 10^-4 | Small decimal |
| 6.022e+23 | - | 6.022 x 10^23 | Avogadro's number |
Quick reference for frequently used numbers in exponential and scientific notation.
Frequently asked questions
What does "exponential form" mean in math?
It depends on context. For whole numbers, exponential form usually means prime factorization written with exponents: 12 = 2^2 x 3. For real numbers, it often means scientific notation: 12,000 = 1.2 x 10^4. In algebra, exponential form of a logarithm means rewriting log_b(c) = a as b^a = c. This calculator handles all three meanings.
How do I convert a logarithm to exponential form?
Use the identity log_b(c) = a is equivalent to b^a = c. Identify the base (b), the result (c), and the exponent (a), then rewrite. Example: log_2(8) = 3 becomes 2^3 = 8. If the base is e (Euler's number), the notation "ln" is used: ln(7.389) = 2 becomes e^2 = 7.389.
What is the prime factorization of a number?
Prime factorization breaks a number down into a product of prime numbers (primes are numbers divisible only by 1 and themselves: 2, 3, 5, 7, 11, ...). When written in exponential form, repeated primes are grouped: 72 = 2 x 2 x 2 x 3 x 3 = 2^3 x 3^2. Every integer greater than 1 has exactly one prime factorization (Fundamental Theorem of Arithmetic).
How do I write a number in scientific notation?
Move the decimal point until you have a number between 1 and 10 (this is the coefficient). Count how many places you moved: if you moved left, the exponent is positive; if you moved right, the exponent is negative. Write the result as coefficient x 10^exponent. Example: 0.00072 - move the decimal 4 places right to get 7.2, so the answer is 7.2 x 10^-4.
Can the exponent in exponential form be negative?
Yes. A negative exponent means the base is divided rather than multiplied: b^-a = 1 / b^a. For example, 2^-3 = 1/8 = 0.125. In scientific notation, a negative exponent of 10 indicates the number is less than 1: 3.5 x 10^-2 = 0.035.
What is the difference between exponential form and standard form?
Standard form just means writing the number the usual way (e.g., 4500). Exponential form uses a base and exponent to express the same value more compactly (e.g., 4.5 x 10^3 in scientific notation, or 2^2 x 5^3 as a prime factorization). For very large or very small numbers, exponential form is far more practical than writing out all the digits.