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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 28: Logarithmic functions differentiation

# Derivative of logarithm for any base (old)

An older video where Sal finds the derivative of log_b(x) (for any base b) using the derivative of ln(x) and the chain rule. Created by Sal Khan.

## Want to join the conversation?

• I dont understand one thing...if (1/ln b) is a constant, shouldnt its derivative be 0 like all the other constants?
• The derivative of all constants IS ZERO. However, if we directly use that property then, the derivative of the entire expression will turn out to be zero, which would be wrong.
Instead, we use another derivative property i.e.,

d/dx [A*f(x)] = A*d/dx [f(x)] , where A = constant

Now, you might be thinking that if we use the aforementioned property, then when or why do we need this following property-

d/dx[A] = 0

We use this property to find the derivative of a polynomial.
For eg.,
d/dx[x^2 + 2] = 2x + 0 = 2x.

Hope you got it.
• Can anyone tell me why and how is log base b of x = ln x/ln b?

• Please read my comment regarding the question financeBoy, I am assuming you 're having the same problem as him. And his question is top rated so I saw his question first.
(1 vote)
• Why is the derivative of lnX = 1/x again?
• Why 1/ln b is a constant?
• 1/ln(b) is a constant because when x changes, the value of that expression stays the same. Its value stays constant.
• Why can you remove a constant from the derivative?
• Because the constant is just a multiplier. It makes no difference when you choose to do the multiplication. Perhaps it would be easier to see with an easier derivative:
Let us find the first derivative of 9x² with respect to x.
We can do this straight up as d/dx (9x²) = (2)(9x) = 18x
Or, we can pull out the constant 9 as in:
d/dx (9x²) = 9 · d/dx (x²) = 9 · (2x) = 18x
• What is the difference between writing "logx" and "lux" ?
• log(x) uses the base 10 while ln(x) called the natural logarithm uses e as the base. The formulas in calculus use the natural logarithm only so the base has to be converted by using the formula Sal mentions at around
Logarithms tell how many time a number(base) has to be raised to for getting another number. Like 10^2 = 100 so log(100) = 2
When log is mentioned without a base it is always 10. The natural logarithm(ln) uses another base called e or Euler's number and is approximately 2.71
• What is the differentiation of log constant with base x with respect to x?
• We have y=log(basex)(c) where c is a constant.
First, we are going to make x be put to both sides.
x^y=c.
next, log both sides. yln(x)=ln(c)
divide by ln(x) y=ln(c)/ln(x)
now, take the derivative of both sides (You need the chain rule for this part which you might not know yet. You can always watch a video on it.). dy/dx=ln(c)/(x*ln(x)^2)
so that's what it is, d/dx( log(basex)(c)) = ln(c)/(x*ln(x)^2)
I hope this helped.
• From onwards, I notice that whenever Sal has to find a derivative of a log with an arbitrary base, he always chooses to represent the log (via change of base formula) as a base-e log and not a base-10 log. Is that how you should always interpret a log with an arbitrary base when finding its derivative?