Derivative of log_x (for any positive base a­1) | AP Calculus AB | Khan Academy

TL;DR

Derivatives of logarithms with arbitrary bases can be found by using the derivative of the natural logarithm.

Transcript

• [Voiceover] We know from previous videos, that the derivative with respect to X of the natural log of X, is equal to 1 over X. What I want to do in this video is use that knowledge that we've seen in other videos to figure out what the derivative with respect to X is of a logarithm of an arbitrary base. So I'm just gonna call that log, base A of ... Read More

Key Insights

• ☺️ The derivative of the natural logarithm of x is 1 over x.
• ⚾ The change of base formula is used to find the derivative of logarithms with arbitrary bases.
• ☺️ The derivative of log base A of x is 1 over the natural logarithm of A times 1 over x.
• ⚾ The change of base formula is useful for calculations involving logarithms with different bases.
• 🧑‍💻 The derivative of log base 10 or log base E can be easily found using the natural logarithm derivative formula.
• 👻 The change of base formula allows for finding logarithms using calculator functions.
• ⌛ The derivative of a constant multiple times a logarithmic function is equal to the constant times the derivative of the logarithm.

Questions & Answers

Q: How can the derivative of a logarithm with an arbitrary base be found?

The derivative can be found using the change of base formula. By rewriting log base A of x as natural log of x divided by natural log of A, the derivative is found to be 1 over the natural log of A times 1 over x.

Q: Why is the change of base formula useful?

The change of base formula is useful because it allows for finding logarithms with different bases using the logarithm function on a calculator, which is typically base 10 or base E (natural log). It saves time and effort in calculation.

Q: How can the derivative of log base 7 of x be found?

The derivative of log base 7 of x is found by applying the formula. The derivative is 1 over the natural log of 7 times 1 over x.

Q: Is the derivative of log base pi of x affected by a constant multiple?

Yes, the derivative of a constant multiple times log base pi of x is found by applying the formula. The derivative is negative 3 over the natural log of pi times x.

Summary & Key Takeaways

• The derivative of the natural logarithm of x is equal to 1 over x.

• To find the derivative of a logarithm with an arbitrary base, the change of base formula is used.

• The derivative of log base A of x is equal to 1 over the natural logarithm of A times 1 over x.