Problem 70 - A Graphical Approach to College Algebra

Question

$$\text { Solve each formula for the indicated variable.}$$ $$p=a+\frac{k}{\ln x}, \text { for } x$$

Step-by-Step Solution

Answer

\(x = e^{\frac{k}{p-a}}\)

Step 1: Isolate the Fraction

Subtract \(a\) from both sides of the equation to get \(p - a = \frac{k}{ ext{ln } x}\).

Step 2: Remove the Fraction

Multiply both sides by \(\ln x\) to eliminate the fraction: \((p - a) \ln x = k\).

Step 3: Solve for \(\ln x\)

Divide both sides by \((p - a)\) to isolate \(\ln x\): \(\ln x = \frac{k}{p - a}\).

Step 4: Solve for \(x\)

Use exponentiation to solve for \(x\): \(x = e^{\frac{k}{p - a}}\).

Key Concepts

Isolate the Variable

In solving logarithmic equations, our main goal is to solve for the variable of interest. This process often begins by isolating the variable, just like we would in any other algebraic equation. When you encounter an equation such as \(p = a + \frac{k}{\ln x}\), to isolate \(x\), we first need to focus on isolating the part of the equation that contains \(x\).

Here, that is where the logarithmic function \(\ln x\) appears within a fraction with \(k\). Therefore, the first step is to eliminate any terms that distract us from this.

• Subtract \(a\) from both sides of the equation. This gives \(p - a = \frac{k}{\ln x}\).
• This effectively isolates the fraction, making \(\ln x\) the main feature that needs further simplification.

This crucial step of isolating the fraction allows us to directly interact with the expression containing \(x\), setting up subsequent steps towards solving the equation entirely.

Solve for the Variable

Once the equation is simplified to \(p - a = \frac{k}{\ln x}\), the next task is to solve for the variable \(x\). This step requires adjusting the current form so that \(\ln x\) stands alone.

How do we accomplish this? We remove the fraction by getting rid of the numerator and denominator configuration. Here's how you can do it:

• Multiply both sides by \(\ln x\) to cancel out the fraction: \((p - a) \ln x = k\).
• Next, divide each side by \(p - a\). This leaves you with \(\ln x = \frac{k}{p - a}\).

This process effectively isolates \(\ln x\), transforming the equation into a simpler format where you can solve directly for \(x\) in the next step. Always ensure to keep track of any multiplication and division steps to avoid errors in calculations.

Exponentiation Technique

After isolating \(\ln x\) in the equation \(\ln x = \frac{k}{p - a}\), the final step is to solve for \(x\) using exponentiation. Exponentiation is a mathematical operation involving raising a number (known as the base) to a power. With logarithmic equations, exponentiation helps translate the logarithmic form back into its original numerical base.

In this case, you will employ the natural base \(e\) (Euler's number approximately 2.71828), as follows:

• Raise \(e\) to the power of each side of the equation: \(x = e^{\frac{k}{p - a}}\).
• This step effectively "undoes" the logarithm, providing the value of \(x\) in its exponential form.

Exponentiation is a powerful tool, especially when dealing with transcendental numbers like \(e\). Remember, this step is crucial in converting a logarithmic expression back into its algebraic identity, allowing you to solve for variables that were initially embedded within a log function.