Problem 122 - College Algebra

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

You will find that \(x = ln(140) / ln(3)\), which simplifies to approximately \(x \approx 4.605\) after calculating on a calculator.

Step 1: Understand the Problem

Given the exponential equation \(3^{x}=140\), the goal is to solve for x. Since both sides are not powers of the same base, a logarithm must be used to find the value of x.

Step 2: Apply Logarithm to Both Sides

To simplify this equation, a logarithm is applied to both sides. The natural logarithm, ln, can be used: \(ln(3^{x})=ln(140)\). Using the properties of logarithms, the x can be brought down in front - this property states that \(ln(a^{b}) = b*ln(a)\). Therefore, the equation simplifies to: \(x*ln(3) = ln(140)\)

Step 3: Isolate the Variable x

After applying the logarithm, isolate the variable x by dividing both sides of the equation by \(ln(3)\). The equation simplifies to: \(x = ln(140) / ln(3)\)

Step 4: Solve the Equation

Finally, solve the equation to find the value of x by dividing \(ln(140)\) by \(ln(3)\) using a calculator.