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Chapter 4: Problem 122

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.

Short Answer

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

Step by step solution

01

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.
02

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 \(x \cdot \ln(3) = \ln(140)\).
03

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.

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