Problem 132 - College Algebra Essentials

Question

If \(f(x)=\log _{b} x,\) show that $$\frac{f(x+h)-f(x)}{h}=\log _{b}\left(1+\frac{h}{x}\right)^{\frac{1}{h}}, h \neq 0.$$

Step-by-Step Solution

Answer

The equation \(\frac{f(x+h)-f(x)}{h}=\log_b\left(1+\frac{h}{x}\right)^{1/h}\) is proved by expressing function's increment in terms of logarithms and then applying the properties of logarithms. This process helps to transform the left hand side of the equation into its right hand side.

Step 1: Express the Increment in Form of Logarithms

The expression in the left hand side, \(\frac{f(x+h)-f(x)}{h}\), represents the increment of the function in terms of \(h\). We start solving by expressing each term inside in terms of logarithms: \(f(x+h)=\log_b(x+h)\) and \(f(x)=\log_b(x)\). Hence, the whole expression becomes \(\frac{\log_b(x+h)-\log_b(x)}{h}\).

Step 2: Apply the Properties of Logarithms

The property \(\log_a m - \log_a n = \log_a \frac{m}{n}\) is applicable in this situation. So the expression becomes: \(\frac{\log_b\left(\frac{x+h}{x}\right)}{h}\). Simplifying this we get \(\frac{\log_b\left(1+\frac{h}{x}\right)}{h}\). By considering \(1/h\) as the new exponent of the logarithm, the expression turns into: \(\log_b\left(1+\frac{h}{x}\right)^{1/h}\).

Step 3 : Final Result

The expression \(\log_b\left(1+\frac{h}{x}\right)^{1/h}\) represents the right hand of the equation, as was required in the exercise.

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential part of mathematical analysis, especially when it comes to studying rates of change and growth processes. In the equation provided, we are dealing with a logarithmic function given by \( f(x) = \log_b x \). Here's what this means:

• The function \( \log_b x \) stands for the logarithm of \( x \) with base \( b \). It represents the power to which the base \( b \) must be raised to get \( x \).
• Logarithmic functions are the inverses of exponential functions. If \( b^y = x \), then \( \log_b x = y \).

Understanding how logarithms work is crucial, as they help simplify mathematical expressions. They excel at transforming multiplicative processes into additive ones, making problems simpler to handle. These transformations are particularly handy when working with problems involving growth rates or time.

Incremental Change

Incremental change refers to the small adjustments or differences in a function's value. In calculus, this concept is key to understanding derivatives and instantaneous rate of change. In our exercise, the expression \( \frac{f(x+h) - f(x)}{h} \) measures the average rate of change of the function \( f \) from \( x \) to \( x+h \):

• The numerator \( f(x+h) - f(x) \) tells us the change in the value of the function as \( x \) increases by \( h \).
• The denominator \( h \) represents the increment in the variable \( x \). It is a small change that approaches zero when analyzing the function's behavior more closely using derivatives.

When \( h \) approaches zero, this expression gives the derivative \( f'(x) \), which tells us how \( f(x) \) changes at an exact point. By understanding this concept, we can calculate the instantaneous rate of change, which is central to calculus.

Properties of Logarithms

Properties of logarithms simplify complex expressions and are a powerful tool in calculus and algebra. These properties allow us to rewrite logarithmic expressions in more manageable forms. In our exercise, one essential property is utilized:

• The difference of logarithms: \( \log_a m - \log_a n = \log_a \frac{m}{n} \). This property allows us to switch from subtraction to division within the logarithmic expression, simplifying calculations significantly.
• The power rule for logarithms: \( \log_a (b^c) = c \cdot \log_a b \). This is useful when dealing with exponents, as shown when we turned the expression into \( \log_b\left(1+\frac{h}{x}\right)^{1/h} \).

By applying these properties, students can more easily manipulate and solve logarithmic expressions. These properties serve as critical tools when working with logarithms in calculus, aiding in differentiation and integration tasks.