How to Change Base Formula for Logarithms?
TL;DR: Most calculators only have two log buttons, log base 10 and ln (base e). So what do you do when the problem asks for log base 7 of something? You change the base. The trick is to divide log of the original number by log of the old base, using whichever base your calculator likes. Suddenly any log in any base becomes calculator-friendly. One little formula, and you stop being limited by what your buttons offer.
Key takeaways:
- Formula: \(\log_b(a) = \dfrac{\log_c(a)}{\log_c(b)}\), where \(c\) is any positive base (usually \(10\) or \(e\)).
- The argument \(a\) goes on top; the original base \(b\) goes on bottom.
- Use \(\log\) (base \(10\)) on the calculator: \(\log_5(20) = \dfrac{\log 20}{\log 5}\).
- Use \(\ln\) (base \(e\)) when working with calculus or exponential growth.
- Alternate form: \(\log_b(a) \cdot \log_c(b) = \log_c(a)\), useful in proofs.
Changing the base formula, as the name suggests, is used to change the base of the logarithm. In this following step-by-step guide, you learn more about the change of base formula.
Changing the base formula, as the name suggests, is used to change the base of the logarithm. You may have noticed that a scientific calculator only has “\(log\)” and “\(ln\)” buttons.
We know that “\(log\) ” stands for a logarithm of base \(10\) and “\(ln\)” stands for a logarithm of base \(e\). But there is no option to calculate the logarithm of a number with bases other than \(10\) and \(e\). Changing the base formula solves this problem.
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A step-by-step guide to the change of base formula for logarithms
The change of base formula is used to write a logarithm of a number with a definite base as the ratio of two logarithms each with the same base that is different from the base of the original logarithm. This is the property of logarithms.
Change of base formula
The change of base formula is:
\(\color{blue}{log _b\left(a\right)=\frac{\left[log _c\left(a\right)\right]}{\left[log _c\left(b\right)\right]}}\)
In this formula:
- The logarithm argument in the numerator is the same as the original logarithm argument.
- The logarithm argument at the denominator is the same as the base of the original logarithm.
- The base of both numerator and denominator logarithms must be the same, and this base can be any positive number other than \(1\).
Note: Another form of this formula is, \(log _b\left(a\right). log _c\left(b\right)=log _c\left(a\right)\), which is also widely used in solving problems.
The Change of Base Formula for Logarithms, Example 1:
Evaluate the value of \(log _{64}\left(8\right)\) using the change of base formula.
Solution:
Apply the base formula change by changing the base to \(10\):
\(log _{64}\left(8\right)=\frac{log\:8}{log\:64}\)
\(=\frac{log\:8}{log\:8^2}\)
\(=\frac{log\:8}{2\:log\:8}\)
\(=\frac{1}{2}\)
Exercises for the Change of Base Formula for Logarithms
Evaluate each logarithm by using the change of base formula.
- \(\color{blue}{log _{\frac{1}{4}}\left(32\sqrt{2}\right)}\)
- \(\color{blue}{log _{16}\left(32\right)}\)
- \(\color{blue}{log _{\frac{1}{27}}\left(243\right)}\)
- \(\color{blue}{log _{0.1}\left(1000\right)}\)
- \(\color{blue}{-\frac{11}{4}}\)
- \(\color{blue}{\frac{5}{4}}\)
- \(\color{blue}{-\frac{5}{3}}\)
- \(\color{blue}{-3}\)
Recommended EffortlessMath Books
For a workbook that covers logarithms inside a full Algebra II program, Algebra II for Beginners walks through log rules, change of base, and exponential equations with worked examples. For test-prep coverage, The Ultimate Algebra Bundle covers logs alongside polynomials, rational functions, and conic sections.
Frequently Asked Questions
What is the change of base formula for logarithms?
The change of base formula says log_b(a) = log_c(a) ÷ log_c(b), where c is any positive base other than 1. The number you are taking the log of goes on top, and the original base goes on the bottom. Most students use c = 10 or c = e, because those are the two log buttons on a calculator.
What is the change of base formula used for?
It lets you evaluate a logarithm in any base using a calculator that only has log (base 10) and ln (base e). It also helps you solve exponential equations, compare logs with different bases, and simplify expressions in proofs. For example, log₇(49) can be rewritten as log 49 ÷ log 7, which equals 2.
How do you evaluate logarithms using the change of base formula?
Rewrite the log as a fraction of common logs, then divide. For log₅(20), compute log 20 ÷ log 5 ≈ 1.3010 ÷ 0.6990 ≈ 1.861. You can check the answer by raising the base to that power: 5^1.861 is very close to 20. Using ln instead of log gives the same result.
How do you solve logarithms using the change of base formula?
When an equation like 3ˣ = 50 has the unknown in the exponent, rewrite it as x = log₃(50). Then use the change of base formula: x = log 50 ÷ log 3 ≈ 1.6990 ÷ 0.4771 ≈ 3.561. Keep several decimal places until the final step so rounding does not throw off your answer.
How do you prove the change of base formula?
Let y = log_b(a), which means bʸ = a. Take log base c of both sides to get log_c(bʸ) = log_c(a). The power rule turns the left side into y·log_c(b). Divide both sides by log_c(b) and you get y = log_c(a) ÷ log_c(b), which is exactly the formula.
Can you use the change of base formula without a calculator?
Yes, if you choose a new base that makes both logs easy. For log₈(4), switch to base 2: log₂(4) ÷ log₂(8) = 2 ÷ 3, so log₈(4) = 2/3. This works well whenever the number and the base are both powers of the same smaller number.
Related EffortlessMath Lessons
If a topic on this page feels rusty, these short lessons go deeper:
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