Question

If f(x)=logx, show that f(x+h)-f(x)/h=log[1+h/x]^1/h, h=/=0 (Picture attached, thank you!)

Answer 1

Answer:

Step by step proof shown below.

Step-by-step explanation:

To prove the equation, you need to apply the Logarithm quotient rule and the Logarithm power rule. Here's how the quotient rule looks like.

$log_b(x/y) = log_b(x) - log_b(y)$

And here's how the power rule looks like

$log_a(x)^n = nlog_a(x)$

First let's apply the quotient rule.

$\frac{f(x+h)-f(x)}{h} = \frac{log_a(x+h)-log_a(x) }{h} = \frac{log_a(\frac{x+h}{x} )}{h}$

Now we can do some quick simplification, and apply the power rule.

$\frac{1}{h} log_a(1 + \frac{h}{x} ) = log_a(1+\frac{h}{x} )^\frac{1}{h}$