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

Step by step solution

01

Understand the logarithmic conversion

To solve an exponential equation where the base on one side can't be converted to the same base on the other side, logarithms can be used. Logarithms are the inverse of exponential functions, which makes them very useful in solving equations of this kind.
02

Apply logarithm to both sides

One needs to apply the log to both sides of the equation. This should look like this: \(\log(3^{x}) = \log(140)\).
03

Apply the Power Rule of Logarithms

With the Power Rule, the exponent of the subject of the log on the left side of the equation can be brought out in front. Therefore, the equation would convert to \(x \cdot \log(3) = \log(140)\).
04

Solve for x

Finally, divide both sides of the equation by \(\log(3)\) to isolate x on one side. Thus, the solution of the equation is \(x = \frac{\log(140)}{\log(3)}\).

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Most popular questions from this chapter

Exercises \(153-155\) will help you prepare for the material covered in the next section. a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=4.5(0.6)^{x}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. As the number of compounding periods increases on a fixed investment, the amount of money in the account over a fixed interval of time will increase without bound.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In \(2006,\) Canada's population exceeded Uganda's by 4.9 million.
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