Home > Calculus > Differentiating Logarithmic Functions without Base e

How do you differentiate #y= log _a x#?

Answer 1

Samantha Adkison

#y' = 1/(xln a)#

the easiest way is to shift the base to e

so #y = log_a x = (log_e x)/(log_e a)# {small demo of what that is so is set out below}

thusly

#y' = 1/x *1/(log_e a) = 1/(xln a)#

the demo

#y = log_a x implies a^y = x# by definition

so we choose to use natural logs because they work so well with calculus

#ln a^y = ln x#

#y ln a = ln x#

#y = ( ln x)/(ln a)#

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Answer 2

Isabella Ackerman

To differentiate ( y = \log_a x ), where ( a ) is a constant base, you can use the following formula:

[ \frac{d}{dx} \log_a x = \frac{1}{x \ln a} ]

This derivative formula holds for any positive constant base ( a ) and for ( x > 0 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.