Problem 122 - Algebra and Trigonometry

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 for \(x\) in the equation \(3^{x}=140\) is approximated at \(x = \frac{ln(140)}{ln(3)}\), which when calculated gives us the value of \(x\) approximately equal to 4.77.

Step 1 Identify the exponential equation

The given exponential equation is \(3^{x}=140\). In this equation, 3 is the base, x is the exponent, and 140 is the result of the exponential operation. The goal is to find the value of x.

Step 2 Take the logarithm of both sides

Applying the logarithm to both sides can help us to isolate variable from exponent. We take natural logarithm (ln) of both sides of the equation. This gives us the equation: \(ln(3^{x}) = ln(140)\).

Step 3 Use the properties of logarithms

We use the property of logarithms that allows us to move the exponent in front of the logarithm. This gives us the equation: \(x*ln(3) = ln(140)\).

Step 4 Isolate the variable

We can now find the value of x by dividing both sides of the equation by \(ln(3)\). This gives us the equation: \(x = \frac{ln(140)}{ln(3)}\).

Step 5 Solve the equation

Solving the equation on the calculator provides the value of x.