Home > Calculus > Differentiating Logarithmic Functions without Base e

How do you differentiate #y= log _3 x#?

Answer 1

#(dy)/(dx)=1/(xln3)#

We can't directly derive logarithms with bases different to #e#, so we have to make this equation in terms of the natural log #ln#.

Using change of base, we can convert this to #y=(lnx)/(ln3)#

#1/(ln3)# is just a constant, and thus will not change.

#(dy)/(dx)=1/(ln3)*1/x->#the derivative of #lnx# is #1/x#

#=1/(xln3)#

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Answer 2

Luke Alvarez

To differentiate ( y = \log_3 x ), you can use the formula for differentiating logarithmic functions. The derivative of ( \log_a x ) with respect to ( x ) is ( \frac{1}{x \ln a} ). Therefore, the derivative of ( y = \log_3 x ) with respect to ( x ) is ( \frac{1}{x \ln 3} ).

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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.