Understanding Logarithmic Derivatives

This video tutorial explains the derivative of logarithms, starting with the natural log and extending to logarithms of arbitrary bases. It introduces the change of base formula, which is essential for converting between different logarithmic bases. The tutorial also demonstrates how to use calculators to find logarithms of various bases and derives the formula for the derivative of a logarithm with an arbitrary base. Finally, it provides examples to illustrate the application of these concepts.

What is the derivative of the natural logarithm of x with respect to x?

• 1/x

• e^x

• x

• ln(x)

What is the change of base formula for converting log base a of b to a different base c?

• log base a of b divided by log base c of a

• log base c of b divided by log base c of a

• log base c of a divided by log base c of b

• log base c of b times log base c of a

Which base is typically used for the log button on calculators?

• Base 10

• Base 16

• Base 2

• Base e

How does the change of base formula help in using calculators for logarithms?

• It allows conversion of any logarithm to base 10 or e.

• It simplifies the logarithm to a single number.

• It eliminates the need for a calculator.

• It changes the logarithm to base 2.

How can you calculate log base 3 of 8 using a calculator with only base 10 logs?

• log(8) / log(3)

• log(8) + log(3)

• log(3) / log(8)

• log(8) * log(3)

What is the derivative of log base a of x in terms of natural logarithms?

• ln(x) / ln(a)

• x / ln(a)

• ln(a) / x

• 1 / (ln(a) * x)

What is the significance of the natural log in finding derivatives of logarithms with arbitrary bases?

• It is used to simplify the expression.

• It complicates the calculation.

• It is the only logarithm with a known derivative.

• It is irrelevant to the process.

If f(x) = log base 7 of x, what is f'(x)?

• 1 / (ln(7) * x)

• ln(7) / x

• x / ln(7)

• ln(x) / ln(7)

What is the derivative of g(x) = -3 * log base pi of x?

• -3 / (ln(pi) * x)

• 3 / (ln(pi) * x)

• -3 * ln(pi) / x

• 3 * ln(pi) / x

What is the role of the constant in the derivative of a logarithmic function with a constant multiplier?

• It is ignored in the derivative.

• It is subtracted from the derivative of the logarithm.

• It is multiplied by the derivative of the logarithm.

• It is added to the derivative of the logarithm.