Problem 42 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the derivative of the function. $$ g(x)=\log _{5} x $$

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = \log_5x\) is \(g'(x) = \frac{1} {x*ln(5)}\)

Step 1: Identify the Expression

We evaluate or simplify:
In Exercises, find the derivative of the function.
$$
g(x)=\log _{5} x
$$

Step 2: Apply Logarithm Properties

Key properties: \(\log_a(a^x) = x\), \(\log_a(1) = 0\), \(e^{\ln x} = x\), \(\ln(e^x) = x\).

Step 3: Result

The derivative of the function \(g(x) = \log_5x\) is \(g'(x) = \frac{1} {x*ln(5)}\)

Key Concepts

Logarithmic Function Basics

Logarithmic functions are essential in mathematics, especially in calculus, because they offer a way to reverse exponential functions. A logarithm is defined as the power to which a base must be raised to obtain a given number. For example, in the expression \( \log_b x \), \( b \) is the base and \( x \) is the number you want to evaluate. The function essentially asks, "To what power must \( b \) be raised, to equal \( x \)?"

When dealing with logarithmic functions, it’s important to understand some specific properties, such as:

• The product rule: \( \log_b(x \cdot y) = \log_b x + \log_b y \)
• The quotient rule: \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• The power rule: \( \log_b(x^c) = c \cdot \log_b x \)

These basic properties allow us to manipulate and simplify logarithmic expressions. When we talk about the derivative of a logarithmic function, we’re often interested in how it changes as \( x \) changes, which leads us into differentiation.

Differentiation of Logarithmic Functions

Differentiation is a fundamental concept in calculus, used to determine the rate at which a function is changing at any point. When finding the derivative of a logarithmic function like \( g(x)=\log _{5} x \), we can use specific differentiation rules for logarithms.

The general formula for finding the derivative of \( \log_b x \) is:
\[\frac{d}{dx} [\log_b x] = \frac{1}{x \cdot \ln(b)}\]
This formula is derived from the chain rule and the change of base formula, which expresses a log of any base in terms of the natural logarithm (\( \ln \)).

Let's see this in practice:

• Given \( g(x)= \log_5(x) \)
• The derivative \( g'(x) \) is \( \frac{1}{x \cdot \ln(5)} \)

This tells us how the function \( g(x) \) changes with respect to \( x \). It is crucial for understanding the behavior and rate of change of logarithmic functions.

Understanding Calculus in Context

Calculus is the branch of mathematics that deals with continuous change. It’s divided mainly into differentiation and integration. While differentiation deals with the rate of change, integration deals with the accumulation of quantities.

In the context of logarithmic functions, calculus allows us to explore deeper applications, such as:

• Modeling real-world phenomena.
• Solving problems involving growth and decay.
• Analyzing data trends through functions and their derivatives.

By applying calculus principles, you can derive insights into how systems evolve over time.

Understanding derivatives, like \( g'(x) = \frac{1}{x \cdot \ln(5)} \), is crucial as it provides the slope of the tangent line at any point \( x \) on the function \( g(x) \). This insight helps in making predictions and understanding the behavior of the function in various situations. With calculus, you unlock a broader perspective on the world governed by mathematical laws.