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

slaggingV 2021-10-19 Answered
If f(x)=logax, show that f(x+h)f(x)h=loga(1+hx)1h,hc0
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Expert Answer

Arham Warner
Answered 2021-10-20 Author has 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.
f(x+h)f(x)h=loga(x+h)logaxh=
=logax+hxh
=1hloga(1+hx)
=loga(1+hx)1h
It is defined for h0  as  1h is not defined.
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