Problem 50 - Essential Calculus Early Transcendentals

Question

Let $$f(x)=\log _{a}\left(3 x^{2}-2\right) .\( For what value of \)a\( is \)f^{\prime}(1)=3 ?$$

Step-by-Step Solution

Answer

The value of \( a \) is \( e^2 \).

Step 1: Find the derivative of the function

Given the function \( f(x) = \log_a(3x^2 - 2) \), we need to find its derivative. The derivative of \( \log_a(u) \) is \( \frac{1}{\ln(a) \cdot u} \cdot u' \). Here, \( u = 3x^2 - 2 \) and \( u' = 6x \). Thus, the derivative of the function, \( f'(x) = \frac{1}{\ln(a) \cdot (3x^2 - 2)} \cdot 6x \).

Step 2: Calculate f'(1) for the given condition

Substitute \( x = 1 \) into the derivative found in Step 1: \[ f'(1) = \frac{6 \cdot 1}{\ln(a) \cdot (3 \cdot 1^2 - 2)} = \frac{6}{\ln(a) \cdot 1} = \frac{6}{\ln(a)}. \] According to the problem, \( f'(1) = 3 \). Therefore, we set the equation \( \frac{6}{\ln(a)} = 3 \).

Step 3: Solve for a

We solve the equation \( \frac{6}{\ln(a)} = 3 \). Multiply both sides by \( \ln(a) \) to get \( 6 = 3 \ln(a) \). Dividing both sides by 3 gives \( \ln(a) = 2 \). We then exponentiate both sides to remove the natural logarithm, so \( a = e^2 \).

Key Concepts

Derivative Calculation

Understanding the calculation of derivatives is essential in solving many calculus problems. In this exercise, we are given a logarithmic function and asked to find its derivative. The function is \( f(x) = \log_a(3x^2 - 2) \). When finding the derivative of a logarithmic function of a general base \( a \), we use the formula:

• \( \frac{1}{\ln(a) \cdot u} \cdot u' \)

Here, \( u \) is the expression inside the logarithm, which in this case is \( 3x^2 - 2 \). The derivative of \( u \) (denoted as \( u' \)) is the derivative of \( 3x^2 - 2 \), which is \( 6x \).
Substituting into the formula for the derivative, we get:

• \( f'(x) = \frac{1}{\ln(a) \cdot (3x^2 - 2)} \cdot 6x \)

This expression gives us the rate of change or slope of our original function \( f(x) \) at any point \( x \). Keeping track of each component in the formula is crucial for accuracy.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithmic function with a base of \( e \), where \( e \approx 2.71828 \). It is important when dealing with logarithmic differentiation, especially when transformations are needed to solve certain equations.
In the solution, we encounter \( \ln(a) \) when finding the derivative. When we set \( f'(1) = 3 \), we have the equation:

• \( \frac{6}{\ln(a)} = 3 \)

By understanding the properties of the natural logarithm, we can manipulate this equation to solve for \( a \). Multiplying both sides by \( \ln(a) \) gives us \( 6 = 3 \ln(a) \), and dividing by 3 leads to \( \ln(a) = 2 \).
The natural logarithm is well-suited for this problem as it provides a straightforward method to manage exponential and growth-related equations.

Exponentiation

Exponentiation is the operation of raising one number, called the base, to the power of another number. When solving the problem, we use exponentiation to find the value of \( a \) once we have \( \ln(a) = 2 \).
To eliminate the natural logarithm, we exponentiate both sides of the equation. This involves using the function \( e^x \), which is the inverse of the natural logarithm. Therefore, exponentiating \( \ln(a) = 2 \) gives us:

• \( a = e^2 \)

Exponentiation transforms the logarithmic equation to its exponential form, allowing us to find \( a \) as a numerical value.
Understanding this concept is important because it helps solve equations involving variable exponents and logarithms.