Problem 22 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The value of \(x\) for which \(f(x) = 3\) is \(x = e^{(\frac{7}{3})}\)

Step 1: Set the function equal to 3

First, we need to set our function equal to three to find the value(s) of \(x\) that make the equation true. So we have the equation \(3 \ln x -4 = 3\).

Step 2: Isolate the logarithmic term

Next, we need to isolate the logarithm on one side of the equation. Solve for \(\ln x\) by adding 4 to both sides then dividing by 3. Now our equation looks like: \(\ln x = \frac{3 + 4}{3} = \frac{7}{3}\).

Step 3: Solve for x

As a final step, we will use the property of logarithms that allows us to write the natural logarithm term as an exponent of the base (which is \(e\) for natural logs). This gives us: \(x = e^{(\frac{7}{3})}\).

Key Concepts

Solving Equations

To solve equations involving a function, we start by equating the given function to the desired outcome, which helps us find the value of the variable that makes the equation true. In this problem, we have the function \(f(x) = 3 \ln x - 4\) and need to find the value of \(x\) when \(f(x) = 3\). This means our first step is to set up the equation \(3 \ln x - 4 = 3\).

Once the equation is set, the next task is to isolate the logarithmic part from the equation, as this brings us closer to solving for the unknown variable. In this instance, \(\ln x\) is our logarithmic term, which we isolate by performing inverse operations: adding or subtracting terms as necessary and eventually dividing or multiplying to get \(\ln x\) alone on one side of the equation.

This process introduces the equation \(\ln x = \frac{7}{3}\), which is ready for solving the exponential form.

Natural Logarithms

Natural logarithms, a type of logarithm with the base \(e\) (Euler's number, approximately 2.718), are fundamental in calculus and many real-world applications like compound interest and population growth. A natural logarithm, denoted as \(\ln\), helps in expressing powers of \(e\).

When we isolate \(\ln x\) in an equation such as \(\ln x = \frac{7}{3}\), we have localized the unknown variable \(x\) under the logarithm, preparing it to be expressed as an exponential equation. This step is crucial, as solving equations involving natural logarithms usually requires converting them into their exponential form \(x = e^{(\ln x)}\).

• Natural logs are used to "undo" exponential functions, making them extremely useful in solving equations where the variable is an exponent.

We express the number \(\frac{7}{3}\) as an exponent of \(e\) to solve for \(x\). This exemplifies switching from logarithmic form to exponential form, which we will explore in the next section.

Exponential Functions

Exponential functions are powerful mathematical tools characterized by the variable being the exponent, like \(x = e^{(\frac{7}{3})}\). These functions grow quite rapidly and are prominent in modeling phenomena that increase at constant ratios like populations and finance.

In the context of solving our equation, after isolating \(\ln x = \frac{7}{3}\), we use the property of logarithms that allows us to express the logarithmic equation as an exponential equation. Here, \(x\) is expressed as \(e^{(\frac{7}{3})}\).

• This transition is key as it allows solving for \(x\) by converting the logarithmic equation into a direct computation.
• Exponential equations simplify calculations and help in finding exact numerical solutions using calculators or software capable of evaluating exponential forms.

This final step produces the value of \(x\), which is \(e^{(\frac{7}{3})}\), concluding our process of solving the function equation effectively.