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# 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:
start fraction, d, divided by, d, x, end fraction, natural log, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, end fraction
start fraction, d, divided by, d, x, end fraction, log, start base, b, end base, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, natural log, left parenthesis, b, right parenthesis, dot, x, end fraction
Notice that natural log, left parenthesis, x, right parenthesis, equals, log, start base, e, end base, left parenthesis, x, right parenthesis is a specific case of the general form log, start base, b, end base, left parenthesis, x, right parenthesis where b, equals, e. Since natural log, left parenthesis, e, right parenthesis, equals, 1 we obtain the same result.
You can actually use the derivative of natural log, left parenthesis, x, right parenthesis (along with the constant multiple rule) to obtain the general derivative of log, start base, b, end base, left parenthesis, x, right parenthesis.

## Practice set 1: argument is $x$x

Problem 1.1
h, left parenthesis, x, right parenthesis, equals, 7, natural log, left parenthesis, x, right parenthesis
h, prime, left parenthesis, x, right parenthesis, equals, question mark

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

## Practice set 2: argument is a polynomial

Problem 2.1
g, left parenthesis, x, right parenthesis, equals, natural log, left parenthesis, 2, x, cubed, plus, 1, right parenthesis
g, prime, left parenthesis, x, right parenthesis, equals, question mark

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

## Want to join the conversation?

• 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.