Change of base formula proof | Logarithms | Algebra II | Khan Academy | Summary and Q&A

TL;DR

The change of base formula for logarithms allows us to calculate logarithms with different bases using logarithms with a known base.

Key Insights

• ⚾ The change of base formula for logarithms allows us to calculate logarithms with any desired base using logarithms with a known base.
• 🟰 By setting logarithm base a of x equal to a variable y, we can rewrite the equation in terms of a to the power of y equals x.
• ⚾ Introducing logarithm base b to both sides of the equation and using logarithm properties, we can derive the change of base formula.
• 🧑‍💻 The formula is particularly useful when our calculators only have functions for natural logarithm or log base 10.

Transcript

What I want to do in this video is prove the change of base formula for logarithms, which tells us-- let me write this-- formula. Which tells us that if I want to figure out the logarithm base a of x, that I can figure this out by taking logarithms with a different base. That this would be equal to the logarithm base b-- so some other base-- base b... Read More

Questions & Answers

Q: What is the purpose of the change of base formula for logarithms?

The change of base formula allows us to calculate logarithms with bases other than what our calculators provide, expanding our ability to compute logarithmic values.

Q: How does the change of base formula work?

The formula states that the logarithm base a of x can be determined by dividing the logarithm base b of x by the logarithm base b of a. It involves using logarithms with a known base to calculate logarithms with a different base.

Q: Why is the change of base formula useful?

The formula is useful when we only have access to calculators with natural logarithm or log base 10 functions. It enables us to calculate logarithms with any desired base using these limited functions.

Q: How can we apply the change of base formula in practice?

To apply the formula, we need to choose a different base, preferably one available on our calculator. We can then calculate the logarithm base b of x and the logarithm base b of a to determine the logarithm base a of x.

Summary & Key Takeaways

• The change of base formula states that the logarithm base a of x can be calculated by taking logarithms with a different base, specifically base b. This formula is useful for calculating logarithms when calculators only have natural logarithm or log base 10 functions.

• By setting logarithm base a of x equal to y, we can rewrite the equation as a to the power of y equals x.

• Introducing logarithm base b to both sides of the equation and using logarithm properties, we can solve for y in terms of logarithm base b, resulting in the change of base formula.