Problem 122 - College Algebra Essentials

Question

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Step-by-Step Solution

Answer

The solution is \(x = \frac{\ln(140)}{\ln(3)}\)

Step 1: Apply natural logarithm to both sides of the equation

Apply the natural logarithm (ln) to both sides of the equation \(3^{x}=140\). This results in the equation \(\ln(3^{x}) = \ln(140)\). A property of logarithm allows us to rewrite the left-hand side of the equation.

Step 2: Use the power rule of logarithms to simplify the equation

The power rule of logarithms states: \(\ln(a^b) = b \cdot \ln(a)\). Applying this rule to the left-hand side gives the equation \(x \cdot \ln(3) = \ln(140)\).

Step 3: Solve for x

Now, to isolate x, divide both sides of the equation by \(\ln(3)\). The resulting equation is \(x = \frac{\ln(140)}{\ln(3)}\). The solution to this equation can be computed using a calculator.