107 LearnersLast updated on August 8th, 2025
Math Formula for Change of Base
In mathematics, the change of base formula is used to convert logarithms from one base to another. This formula is particularly useful when calculating logarithms in bases not supported by standard calculators. In this topic, we will learn the formula for changing the base of logarithms.
List of Math Formulas for Change of Base
The change of base formula is an essential tool in logarithms. Let’s learn the formula to convert logarithms from one base to another.
Math Formula for Change of Base
The change of base formula is used to rewrite a logarithm to a different base. It is calculated using the formula: If you have log base b of a number x, it can be expressed as:
log_b(x) = log_c(x) / log_c(b)
where c is the new base you want to convert to, and log_c denotes the logarithm to base c.
Importance of Change of Base Formula
The change of base formula is crucial in mathematics as it allows the calculation of logarithms with any base using a calculator that typically supports only base 10 (common logarithm) or base e (natural logarithm). By understanding this formula, students can easily work with logarithmic expressions in different bases, which is particularly useful in algebra and calculus.
Tips and Tricks to Memorize Change of Base Formula
Students often find logarithms challenging. Here are some tips and tricks to master the change of base formula:
- Remember the formula as "log old base divided by log new base."
- Practice converting logarithms between common bases like 10 and e to become comfortable with the process.
- Use flashcards to memorize the formula and rewrite it for a quick recall, and create a formula chart for a quick reference.
Real-Life Applications of Change of Base Formula
In real life, the change of base formula is used in various fields where logarithmic calculations are required. Here are some applications of the change of base formula:
- In computer science, to convert between different logarithmic time complexities.
- In finance, to calculate compound interest over different periods.
- In engineering, to solve equations involving exponential growth or decay.
Common Mistakes and How to Avoid Them While Using Change of Base Formula
Students make errors when using the change of base formula. Here are some mistakes and the ways to avoid them, to master the formula.
Mistake 1
Using Incorrect Base in Conversion
Students sometimes choose the wrong base for conversion. To avoid this error, always ensure that you are converting to a base that your calculator supports, such as base 10 or base e.
Mistake 2
Forgetting to Divide by the Logarithm of the Original Base
A common mistake is to forget to divide by the logarithm of the original base. Always remember that log_b(x) = log_c(x) / log_c(b) requires both parts.
Mistake 3
Confusing Bases
Students often confuse which base they are converting from and which base they are converting to. Clearly label your original and new bases to avoid confusion.
Mistake 4
Misapplying the Formula
Misapplying the formula occurs when students do not follow the proper order of operations. Ensure that you correctly calculate the numerator and the denominator separately before dividing.
Examples of Problems Using Change of Base Formula
Problem 1
Convert log base 2 of 8 to base 10?
The conversion gives 3
Explanation
Using the change of base formula:
log_2(8) = log_10(8) / log_10(2)
Calculator gives: log_10(8) ≈ 0.9031 and log_10(2) ≈ 0.3010
Thus, log_2(8) = 0.9031 / 0.3010 ≈ 3
Problem 2
Convert log base 5 of 25 to base e?
The conversion gives 2
Explanation
Using the change of base formula:
log_5(25) = log_e(25) / log_e(5)
Calculator gives: ln(25) ≈ 3.2189 and ln(5) ≈ 1.6094
Thus, log_5(25) = 3.2189 / 1.6094 ≈ 2
Problem 3
Convert log base 4 of 64 to base 10?
The conversion gives 3
Explanation
Using the change of base formula:
log_4(64) = log_10(64) / log_10(4)
Calculator gives: log_10(64) ≈ 1.8062 and log_10(4) ≈ 0.6021
Thus, log_4(64) = 1.8062 / 0.6021 ≈ 3
Problem 4
Convert log base 3 of 9 to base e?
The conversion gives 2
Explanation
Using the change of base formula:
log_3(9) = log_e(9) / log_e(3)
Calculator gives: ln(9) ≈ 2.1972 and ln(3) ≈ 1.0986
Thus, log_3(9) = 2.1972 / 1.0986 ≈ 2
Problem 5
Convert log base 7 of 49 to base 10?
The conversion gives 2
Explanation
Using the change of base formula:
log_7(49) = log_10(49) / log_10(7)
Calculator gives: log_10(49) ≈ 1.6902 and log_10(7) ≈ 0.8451
Thus, log_7(49) = 1.6902 / 0.8451 ≈ 2
FAQs on Change of Base Formula
1.What is the change of base formula?
2.Why do we use the change of base formula?
3.How do you convert log base 10 to base e?
4.Can the change of base formula be used for any bases?
5.What is the common mistake when using the change of base formula?
Glossary for Change of Base Formula
- Logarithm: A mathematical operation that determines the power to which a base number must be raised to obtain a particular value.
- Base: The number that is raised to a power in a logarithm.
- Natural Logarithm: A logarithm with base e, where e is approximately 2.718.
- Common Logarithm: A logarithm with base 10.
- Change of Base Formula: A formula used to convert a logarithm from one base to another.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
