Problem 64 - College Algebra

Question

For the following exercises, solve the equation for \(x\), if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. $$ \ln (x)-\ln (x+3)=\ln (6) $$

Step-by-Step Solution

Answer

No valid solution exists, as the result does not satisfy the domain conditions.

Step 1: Use Properties of Logarithms

Combine the logarithms on the left side using the quotient rule for logarithms: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\). The equation becomes \(\ln\left(\frac{x}{x+3}\right) = \ln(6)\).

Step 2: Exponentiate Both Sides

Remove the logarithm by exponentiating both sides: If \(\ln(a) = \ln(b)\), then \(a = b\). Applying this, we get \(\frac{x}{x+3} = 6\).

Step 3: Solve the Rational Equation

Isolate \(x\) by cross-multiplying to clear the fraction: \(x = 6(x + 3)\). Simplify this to \(x = 6x + 18\).

Step 4: Rearrange and Solve for x

Move all terms involving \(x\) to one side: \(x - 6x = 18\). This simplifies to \(-5x = 18\). Divide both sides by \(-5\) to solve for \(x\): \(x = -\frac{18}{5}\).

Step 5: Verify the Solution

Since \(x\) must be greater than \(-3\) to keep the logarithm defined (logarithms of non-positive numbers are undefined), check \(x = -\frac{18}{5} < -3\). This indicates no valid intersection exists within the domain of logarithm functions for this solution.

Key Concepts

Logarithmic Equations

Logarithmic equations involve variables inside the logarithm function. They can initially appear complex, but with the proper techniques, you can solve them efficiently.
To solve a logarithmic equation like \( \ln(x) - \ln(x + 3) = \ln(6) \), you'll often use properties such as combining logs or changing their form.

A key step in solving this type of equation is to recognize opportunities to use logarithmic properties, such as the quotient rule, which simplifies the difference into a single logarithm:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^p) = p\log_b(M) \)

After combining the logarithms, the equation is easier to work with, since exponentiating both sides can eliminate the logs altogether and convert them into a more familiar algebraic form. This is crucial in progressing towards a solution.

Properties of Logarithms

Understanding the properties of logarithms is critical in both simplifying equations and making them solvable. Logarithmic properties allow you to manipulate complex expressions into manageable forms.

The main properties include:

• Quotient Rule: As in our example, \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). This helps combine multiple logarithmic terms into one.
• Product Rule: Similar to the quotient rule, this is useful when you have a sum of logarithms.
• Power Rule: \( \ln(x^n) = n\ln(x) \), which is perfect for pulling exponents out of logarithms.
• Change of Base Formula: This is useful for converting a logarithm to a different base: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \).

Knowing when and how to apply these rules transitions logarithmic equations toward solutions and makes manipulating such expressions straightforward.
By combining these techniques, as in the original example, you can often drastically simplify equations and solve for the variable.

Graphing Equations

Graphing logarithmic equations can reveal solutions that are not immediately clear from algebraic manipulation alone.
The process of graphing involves plotting each side of the equation and observing the intersection points.

In our example, after transforming the equation, you could graph \( y = \ln\left(\frac{x}{x+3}\right) \) and \( y = \ln(6) \).

• An intersection point on these graphs would indicate a solution because it shows equal output for both the left and right sides of the equation for that \( x \) value.
• In cases where no intersection occurs, as with our \( x < -3 \) constraint, the solution is invalid. This highlights how graphing identifies domain issues.

Graphing gives a visual perspective of logarithmic behavior and serves as a verification tool for solutions derived from algebraic methods.
It's a vital skill in examining the function’s nature and discovering physical meaning behind abstract calculations.