Main content

## Calculus, all content (2017 edition)

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

Lesson 28: Logarithmic functions differentiation- Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
- Derivative of logₐx (for any positive base a≠1)
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Differentiate logarithmic functions
- Differentiating logarithmic functions using log properties
- Derivative of logarithm for any base (old)
- Differentiating logarithmic functions review

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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?(22 votes)
- 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.(40 votes)

- Can anyone tell me why and how is log base b of x = ln x/ln b?

Thanks for your help!(3 votes)- 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?(6 votes)
- Why 1/ln b is a constant?(3 votes)
- 1/ln(b) is a constant because when x changes, the value of that expression stays the same. Its value stays
**constant**.(11 votes)

- Why can you remove a constant from the derivative?(3 votes)
- 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(8 votes)

- What is the difference between writing "logx" and "lux" ?(2 votes)
- 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 around0:50

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(8 votes)

- What is the differentiation of log constant with base x with respect to x?(2 votes)
- 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.(3 votes)

- From0:40onwards, 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?(2 votes)
- It is not about interpreting (what ever you meant by that). The only reason, we transform to base-e is simply that we know its derivative. There is (as far as I know) no other way to do it.(2 votes)

- Why EXACTLY can we take 1/lnb out of the derivative? inb4 go see some proof that is much farther down the line of lecture videos. you should probably prove things before you start using them, or people will get lost.(2 votes)
- Yes, any constant can be factored out of a derivative. There is no actual need to do that, but sometimes it is more convenient to do so.(2 votes)

- what if there is a constant in front of the log(2 votes)

## Video transcript

We already know that the
derivative with respect to x of the natural log
of x is equal to 1/x. But what about the
derivative, not of the natural log of
x, but some logarithm with a different base? So maybe you could
write log base b of x where b is
an arbitrary base. How do we evaluate
this right over here? And the trick is
to write this using the change of base formula. So we could write it
in terms of logarithms. We know that log--
I'm just going to restate the change
of base formula. And I'm going to change
from log base b to log base e, which is essentially
the natural log. So the change of
base formula, we prove it elsewhere on the site. Feel free to search for
it on the Khan Academy. The change of base
formula tells us that log base b of x is
equal to the natural log, if we want to go to log base e. The natural log of
x over the natural-- actually let me write it
as an explicit logarithm so it makes it clear
what I'm doing. Log base e of x over
log base e of b, which is the exact same
thing as the natural log of x over the natural log of b. So all we have to do
is rewrite this thing. This is equal to the
derivative with respect to x of the natural log of
x over the natural log of b. Or we could even write it
as 1 over the natural log of b times the natural log of x. And now this becomes
pretty straightforward. Because what we
have right here, 1 over the natural log of
b, this is just a constant that's multiplying
the natural log of x. So we could take it
out of the derivative. So this is the same thing
as 1 over the natural log of b times the
derivative with respect to x of the natural log of x. And we know what
to do with this. This thing right over here is
just going to be equal to 1/x. So we end up with 1 over the
natural log of b times 1/x. So we end up with 1
over the natural log of b times 1/x, or 1
over the natural log of b, which is just
a number times x. So if someone asks you what
is the derivative with respect to x of log base 5 of
x, well, now you know. It's going to be 1 over
the natural log of 5 times x, just like that.