# Derivative of log_x (for any positive base aÃÂ­1) | AP Calculus AB | Khan Academy | Summary and Q&A

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July 22, 2016
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Derivative of log_x (for any positive base aÃÂ­1) | AP Calculus AB | Khan Academy

## TL;DR

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

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### 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.