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Change Of Base Formula Calculator

\[\log_a(X) = \frac{\log_b(X)}{\log_b(a)}\]


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Sometimes calculating a logarithm can be challenging, especially when the numbers are not compatible with the base of the logarithmic function. That's where the change of base formula calculator comes in to make things easier for you.

This tool helps you to convert logarithms from one base to another and make calculations more manageable when working with logarithms in different bases.

What is a logarithm?

“In mathematics, the logarithm is the inverse operation of exponentiation.”

 The logarithm is also called a log, it tells you the power (exponent) to which a base needs to be raised in order to equal a given number.

How do I change the base of a log?

To change the base of a logarithm, you can use the change of base formula. The formula is as follows:

\(\log_b a = \frac{\log_c a}{\log_c b}\)


  • \(log_b a\) – This represents the logarithm of “a” to the base “b”
  • \(log_c a\) – This is the logarithm of “a” to the base “c”
  • \(log_c b\) – This stands for the logarithm of “b” to the base “c”

So, to change the base from (b) to (c), you have to take the logarithm of (a) to the base (c) and divide it by the logarithm of (b) to the base (c).

Here is a step-by-step procedure:

  • 1. Identify the original logarithm you want to change, let’s say \(log_b a\)
  • 2. Apply the change of base formula
  • 3. Choose the new base (c) that you want for your logarithm


⁡Suppose we want to find the value of (\log_4 32) using the change of base formula. Here’s how we do it step by step:

  1. Given:
  • Base: (b = 4)
  • Number: (a = 32)
  • New base: (c = 2) (we will use base 2 for the change)
  1. Apply the Change of Base Formula: 
  • \[ \log_4 32 = \frac{\log_2 32}{\log_2 4} \]
  1. Calculate the Individual Logarithms:
  • \(\log_2 32\): Evaluate this using your calculator. It turns out to be approximately 5
  • \(\log_2 4\): This is simply 2, because \(2^2 = 4\)
  1. Combine the Results: 
  • \[ \frac{\log_2 32}{\log_2 4} = \frac{5}{2} = 2.5 \]

So, \(\log_4 32 \approx 2.5\), you can also calculate this by using this change of base formula calculator provided by Calculatored.

Logarithm Conversion Examples:

Original Logarithm New Base Change of Base Formula Solution
\(log_3(8)\) 5 \(log_5(8) = log_3(8) / log_3(5)\) ≈ 4.28
\(log_2(16)\) 10 \(log_10(16) = log_2(16) / log_2(10)\) 4
\(ln(4)\) 3 \(log_3(4) = ln(4) / ln(3)\) ≈ 1.26
\(log_e(2)\) 2 \(log_2(e) = log_e(2) / log_e(2)\) 1

How to use our change of base formula calculator?

  • Simply enter the values for a,b, and x in the given section
  • Hit the Calculate button
  • The calculator will give you the results for your Log Base Conversions along with the step-by-step solution


What is a change of base?

A change of base refers to the process of converting a logarithm from one base to another using a specific formula called the "Change of Base Formula.

Shaun Murphy

I'm graduated in biomedical and electrical engineering. Specialization in sports and medical topics but will gladly tackle everything you throw at him. He is a sailor, hiker, and motorcyclist in his free time.

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