proving ALL logarithm properties using calculus | Summary and Q&A

TL;DR

The video proves logarithm properties by showing that the derivative of the natural logarithm of x is 1/x, and then using this result to prove properties such as the power rule and the product rule.

Key Insights

• 😰 The natural logarithm of x can be defined as the integral from 1 to x of 1/t dt.
• ☺️ The derivative of the natural logarithm of x is 1/x, which can be derived using the first fundamental theorem of calculus.
• 📏 The power rule for logarithms, ln(x^r) = rln(x), and the product rule for logarithms, ln(xy) = ln(x) + ln(y), can be proven by differentiating the functions and applying the derivative result.

Transcript

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Questions & Answers

Q: How is the natural logarithm of x defined?

The natural logarithm of x is defined as the integral from 1 to x of 1/t dt.

Q: How is the derivative of the natural logarithm of x calculated?

The derivative of the natural logarithm of x is calculated using the first fundamental theorem of calculus, which states that the derivative of an integral function is equal to the function itself evaluated at the upper limit. In this case, the derivative is 1/x.

Q: How can the power rule for logarithms be proven using the integral definition?

The power rule for logarithms, which states that ln(x^r) = rln(x), can be proven by taking the derivative of ln(x^r) using the integral definition and observing that it is equal to r/x, which matches the result expected from the power rule.

Q: How can the product rule for logarithms be proven using the integral definition?

The product rule for logarithms, which states that ln(xy) = ln(x) + ln(y), can be proven by differentiating ln(xy) using the integral definition and showing that it is equal to 1/x + 1/y, which matches the result expected from the product rule.

Summary & Key Takeaways

• The video explains that the natural logarithm of x can be defined as the integral from 1 to x of 1/t dt.

• It shows how to find the derivative of the natural logarithm of x using the first fundamental theorem of calculus, which gives the result of 1/x.

• The video then proves properties of logarithms, such as the power rule (ln(x^r) = rln(x)) and the product rule (ln(xy) = ln(x) + ln(y)), by differentiating the functions and applying the derivative result.