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Exponential Form Calculator

Created by Krishna Nelaturu
Reviewed by Rijk de Wet
Last updated: Jan 18, 2024


Whether you wish to write an integer in exponential form or convert a number from log to exponential format, our exponential form calculator can help you.

In mathematics, we say a number is "in exponential form" when one or more exponents are involved. In the article below, we will briefly discuss the following fundamentals to ease you into the subject further:

  • Writing integers in exponential form;
  • Calculating log to exponential form conversion; and
  • Calculating exponential to logarithm form.

Writing integers in exponential form

An integer can be expressed as a product of primes through prime factorization. For example, 250 can be written as:

250=2×5×5×5250 = 2 \times 5 \times 5 \times 5

We can use the exponential form to express this more concisely:

250=2×53250 = 2 \times 5^3

We say that 2×53 2 \times 5^3 is the exponential form of 250250.

Writing the number in the exponential form retains the vital information (the prime factors) while saving space. If you want to learn how to prime factorize a number, head to our prime factorization calculator.

Since we depend on prime factorizing to write a number this way, we can only express non-zero whole numbers in the exponential form. And of course, since prime numbers' only factors are themselves, their prime factorizations are themselves, too!

Exponential form and exponential notation are different. Exponential notation is similar to the exponential form but doesn't rely on prime factorization. Exponential notation is a way to represent any number in a more accessible format for calculations. Head to our exponential notation calculator to learn more.

Calculating log to exponential form conversion

A logarithmic number can be converted into its corresponding exponential form due to their relation:

ba=c  logbc=aea=c  lnc=ab^a = c \ \Longrightarrow \ \log_b c = a\\[0.5em] e^a = c \ \Longrightarrow \ \ln c = a\\

where:

  • bb — The base;
  • aa — The exponent;
  • cc — A number; and
  • ee — Euler's number.

🙋 To learn more about logarithms, visit our log calculator.

For example, consider ln15=2.71\ln 15 = 2.71. We can convert to exponential form by raising ee to both sides:

ln152.71eln15=e2.7115=e2.71\begin{align*} \ln 15 &\approx 2.71\\ \Rightarrow e^{\ln 15} &= e^{2.71}\\ 15 &= e^{2.71} \end{align*}

Consider another example, log28=3\log_2 8 = 3, which we can convert by raising 22 to both sides:

log28=32log28=238=23\begin{align*} \log_2 8 &= 3\\ \Rightarrow 2^{\log_2 8} &= 2^{3}\\ 8 &= 2^{3} \end{align*}

Calculating exponential to logarithm form

Calculating exponential to logarithm form is basically the inverse of what we did in the last section.

logbc=aba=clnc=aea=c\begin{aligned} \log_b c &= a \Longrightarrow b^a = c \\[0.5em] \ln c &= a \Longrightarrow e^a = c\\ \end{aligned}

For example, consider 25=322^5 = 32, which we can convert by applying log2\log_2 on both sides:

25=32log2(25)=log2325=log232\begin{align*} 2^5 &= 32\\ \Rightarrow \log_2(2^5) &= \log_232\\ 5 &= \log_2 32 \end{align*}

How to use this exponential form calculator

Our exponential form calculator is straightforward to use:

  1. Select what conversion you wish to perform. You can ask our calculator to:

    • Write the exponential form of a whole number;
    • Convert log to exponential form; or
    • Convert exponential to log form.
  2. Enter the whole number you wish to convert to exponential form, and our calculator will do the rest.

  3. To convert from log form to exponential form (logbc=aba=c\log_bc =a \Longrightarrow b^a=c):

    • Enter the logarithm base (b) along with the number (c). The calculator converts it to exponential form and gives you the result.

    • Tip: If base b=eb = e, enter e in the field.

  4. To convert from exponential form to log form (ba=clogbc=ab^a=c \Longrightarrow \log_bc = a) to log form:

    • Enter the base (b) along with the exponent (a). Our tool will calculate the logarithm form from the exponential form.

    • Tip: If base b=eb = e, enter e in the field.

FAQ

What is the exponential form of 128?

The exponential form of 128 = 27. To find this answer, follow these steps:

  1. Prime factorize the number: 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2.
  2. Find the exponent of the prime factor 2. Since 2 is multiplied 7 times, the exponent is 7.
  3. Write the number in the form factorexponent to get: 128 = 27
  4. Verify the result with our exponential form calculator.

How do you write 3×3×3×3 in the exponential form?

The exponential form of 3×3×3×3 = 34. To arrive at this answer, follow these simple steps:

  1. Count the number of times the prime factor 3 is multiplied with itself to get the exponent 4.
  2. Write the number in the form factorexponent to get: 34.
  3. Verify with our exponential form calculator.

Can I write 24.65 in the exponential form?

You cannot write 24.65 in the exponential form since it is not a whole number. You can, however, write it in the exponential notation 2.465 × 101, which is different from the exponential form. To learn how to do this, visit our exponential notation calculator.

What is the relationship between logarithm and exponential functions?

A logarithm and an exponential function with the same base are inverse functions of each other. If ba = c, then logb(c) = a, and vice versa.

Krishna Nelaturu
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