Problem 119 - Introductory and Intermediate Algebra for College Students 4th

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 to the given equation \(3^{x}=140\) is \( x = \frac{\log 140}{\log 3}\).

Step 1: Translate the equation to logarithmic form

Taking log base 3 for both sides of the equation \(3^{x}=140\). It can be translated to \(x = \log_{3}{140}\)

Step 2: Use the change of base formula

The change of base formula is defined as \(\log_b a = \frac{\log a}{\log b}\). Using this formula, the equation \(x = \log_{3}{140}\) can be rewritten as \(x = \frac{\log 140}{\log 3}\)

Step 3: Calculate the value

Now that the problem has been simplified further into a more manageable form, calculate the value by dividing \(\log(140)\) by \(\log(3)\). This can be done on any scientific calculator.