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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q 104. (page 309)

Difference Quotient If $f (x) = \log_{a} x$ , show that $\frac{f (x + h) - f (x)}{h} = \log_{a} {(1 + \frac{h}{x})}^{\frac{1}{h}}, h 鈮� 0$ .

Short Answer

Expert verified

The result is shown in the explanation.

Step by step solution

01

Step 1. Given information.

Consider the given function.

$f (x) = \log_{a} x$

02

Step 2. Calculation.

Substitute the values of the function on the left side of the equation.

$\begin{array}{rcl} \frac{f (x + h) - f (x)}{h} & = & \frac{\log_{a} (x + h) - \log_{a} x}{h} \\ = & \frac{1}{h} \log_{a} (\frac{x + h}{x}) \\ = & \frac{1}{h} \log_{a} (1 + \frac{h}{x}) \\ = & \log_{a} {(1 + \frac{h}{x})}^{\frac{1}{h}} \end{array}$

03

Step 3. Conclusion.

Since the left side is equal to the right side.
The given equation has been proved.

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