Difference Quotient \text{ If } f(x)=\log_ax,\text{ show that }\frac{f(x+

shadsiei

shadsiei

Answered question

2021-10-20

Difference Quotient
  If  f(x)=logax,  show that  f(x+h)f(h)h=loga(1+hx)1h,h0

Answer & Explanation

rogreenhoxa8

rogreenhoxa8

Skilled2021-10-21Added 109 answers

Step 1
For a function f the difference quotient f(x+h)f(x)h is used to calculate the derivative by taking the limit 0. To compute difference quotient for given functions properties of logarithms will be used.
Exponent property of logarithms is ylogx=logxy. Difference property of logarithms is logxlogy=log{xy} These properties are true for logarithm of any base.
Step 2
Given function is f(x)=logax. Use the properties of logarithm given in step 1 to prove the given statement.
f(x+h)f(x)h=loga(x+h)logxh
=logax+hxh
1hloga(1+hx)
=loga(1+hx)1h
Hence, proved.

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