If f(x)=\log_ax, show that \frac{f(x+h)-f(x)}{h}=\log_a\left(1+\frac{h}



Answered question


If f(x)=logax, show that f(x+h)f(x)h=loga(1+hx)1h,hc0

Answer & Explanation

Arham Warner

Arham Warner

Skilled2021-10-20Added 102 answers

Step 1
The difference quotient f(x+h)f(x)h can be used to calculate the first derivative of the function by taking the limit h0. It will give f'(x) if it exists.
For the given problem, two properties of logarithms will be used. The first is the difference property, which is logaxlogay=logaxy. The second one is the exponent property of logarithms, which is ylogax=logaxy
Step 2
Substitute f(x)=logax in the difference quotient f(x+h)f(x)h and simplify using the properties of logarithms.
It is defined for h0  as  1h is not defined.

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