Problem 21 - Precalculus : Building Concepts and Connections

Question

Use \(f(x)=3 \ln x-4\). For what value of \(x\) will \(f(x)=2 ?\)

Step-by-Step Solution

Answer

The value of \(x\) for which \(f(x) = 2\) is \(x = e^2\)

Step 1: Write Down the Function

Start by writing down the given function: \(f(x) = 3 \ln x - 4\)

Step 2: Plug in the Given Value

The problem asks at what value of \(x\) will \(f(x)=2\). Therefore, substitute \(f(x)\) by 2: \(2 = 3 \ln x - 4\)

Step 3: Solve for \(x\)

Add 4 to both sides of the equation to isolate the logarithm function: \(2 + 4 = 3 \ln x\). Then, to isolate \( \ln x\), divide both sides by 3. You obtain \( \ln x = 2\). Utilizing the property of the logarithm that says that \(e^{\ln a} = a\), you can find the value of \(x\) by substituting \(\ln x\) with \(2\): \(x = e^2\)

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a fundamental concept in mathematics associated with exponential growth and decay. It is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm function, \(\ln(x)\), is defined only for positive real values of \(x\). This function is essential because it can transform multiplicative processes into additive ones, simplifying calculations.One key property of the natural logarithm is:

• \( \ln(e) = 1 \) because any number raised to the power of 1 is itself, hence \(e^1 = e\).
• The inverse relationship between the natural logarithm and the exponential function is expressed as \( e^{\ln(a)} = a \), which is useful in solving equations involving \(\ln \).

Understanding the natural logarithm function is crucial when working with equations that model real-world phenomena, like finance and population growth, using exponential functions.

Solving Equations

Solving equations involves finding the values of variables that satisfy a mathematical statement. In the given problem, we aim to find the value of \(x\) for which \(f(x) = 2\) given that \(f(x) = 3 \ln x - 4\).Here's a breakdown of the process:

• Start by equating \(f(x)\) with 2: \(2 = 3 \ln x - 4\).
• To isolate \(\ln x\), add 4 to both sides: \(2 + 4 = 3 \ln x\) simplifies to \(6 = 3 \ln x\).
• Next, divide both sides by 3 to solve for \(\ln x\): \( \ln x = 2 \).

By solving for \(\ln x\), we have determined what \(x\) makes the equation true. This systematic approach can be applied to solve a variety of logarithmic equations.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a e^{bx} \), where \(e\) is Euler's number, \(a\) and \(b\) are constants. They describe processes that change at rates proportional to the current value, often appearing in natural processes, such as growth and decay.A key relationship used in logarithmic equations is the conversion between logarithms and exponentials:

• If \( \ln x = y \), it can be rewritten in exponential form as \( x = e^y \).

In our specific problem, after solving \( \ln x = 2 \), the use of exponential functions allows us to express \(x\) using the exponent: \(x = e^2\). Therefore, \(x\) is approximately 7.389, showing how interrelated logarithmic and exponential functions are. Recognizing these patterns and conversions is crucial in solving complex mathematical problems involving these functions.