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### Course: Class 12 math (India)>Unit 5

Lesson 15: Logarithmic functions differentiation

# Differentiating logarithmic functions review

Review your logarithmic function differentiation skills and use them to solve problems.

## How do I differentiate logarithmic functions?

First, you should know the derivatives for the basic logarithmic functions:
$\frac{d}{dx}\mathrm{ln}\left(x\right)=\frac{1}{x}$
$\frac{d}{dx}{\mathrm{log}}_{b}\left(x\right)=\frac{1}{\mathrm{ln}\left(b\right)\cdot x}$
Notice that $\mathrm{ln}\left(x\right)={\mathrm{log}}_{e}\left(x\right)$ is a specific case of the general form ${\mathrm{log}}_{b}\left(x\right)$ where $b=e$. Since $\mathrm{ln}\left(e\right)=1$ we obtain the same result.
You can actually use the derivative of $\mathrm{ln}\left(x\right)$ (along with the constant multiple rule) to obtain the general derivative of ${\mathrm{log}}_{b}\left(x\right)$.
Want to learn more about differentiating logarithmic functions? Check out this video.

## Practice set 1: argument is $x$‍

Problem 1.1
$h\left(x\right)=7\mathrm{ln}\left(x\right)$
${h}^{\prime }\left(x\right)=?$

Want to try more problems like this? Check out this exercise.

## Practice set 2: argument is a polynomial

Problem 2.1
$g\left(x\right)=\mathrm{ln}\left(2{x}^{3}+1\right)$
${g}^{\prime }\left(x\right)=?$

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• I can follow these equation but i can't follow x^x any advice? Using logarithmic differentiation.
• From my understanding, you'd like help with how to differentiate x^x. This is how you do it:

y=x^x

Take the logs of both sides:
ln(y) = ln(x^x)

Rule of logarithms says you can move a power to multiply the log:
ln(y) = xln(x)

Now, differentiate using implicit differentiation for ln(y) and product rule for xln(x):
1/y dy/dx = 1*ln(x) + x(1/x)
1/y dy/dx = ln(x) + 1

Move the y to the other side:
dy/dx = y (ln(x) + 1)

But you already know what y is... it is x^x, your original function. So sub in:
dy/dx = x^x(ln(x) + 1)

And you're done.
• I have a natural logarithm with e^x/1+e^x. I separated it with the log rules but then I'm stuck. Any advice?
• i think you are asking about finding d/dx( ln( e^x / 1 + e^x) ). so im solving for that and here it is:
we can write ==> ln(e^x / 1+e^x) as ln(e^x) - ln(1+e^x)

so now when we differentiate we can differentiate them independently.
so d/dx( ln( e^x / 1 + e^x) ) = d/dx( ln(e^x) ) - d/dx( ln(1+e^x) )
= ( (1/e^x) *e^x ) -( ( 1/(1+e^x) ) * e^x )

let me know if we have any confusion.
• can this statement be true?

2 log y = log y^2
(1 vote)
• Are there “rules” for when you can(not) use logarithmic differentiation (including implicit)?

I ask because of the following KA problem: “Find dy/dx for x=√(xy+1)” For that problem I attempted to immediately use logarithmic differentiation, e.g. ln(x)=ln(√(XY+1)).

However having now worked on it a good deal I have come to understand that logarithmic differentiation generates an incorrect result.

Why doesn’t logarithmic differentiation work in this case? (I speculate that perhaps it is because there is a single term that has more than one variable – e.g. XY messes it up – but that is just a guess).

Note that the following answer is not sufficient: “You shouldn’t use logarithmic differentiation on that problem.” E.g. I (now) understand it won’t work - I want to know WHY it doesn’t work - what is the rule I should use so that I don't try to do that again in the future? 😉.

Thanks,
kevin
• Logarithmic differentiation should work here. Can you provide us with your steps so we can perhaps find an error?
• What do you do if the "x" is not simply x, but is raised to a power or if the equation is log base 4 of x-2?
(1 vote)
• 1/x(ln(a)) or I could do natural log of y equals (the power- assuming that the power is a variable) times the natural log of x. If the power is a number I would multiply it by the coefficient of x and subtract 1 from the exponent.