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Chapter 12: Problem 118

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Short Answer

Expert verified
To solve an exponential equation where both sides have the same base, rewrite both sides so that they are expressed as powers of this base, set the exponents equal to each other, and then solve the resulting equation.

Step by step solution

01

Express Both Sides as Powers of the Same Base

To begin, it is important to express both sides of the given equation as powers of the same base. Identify the common base that both terms of the equation can be raised to in order to express them equivalently. If the base terms of the equation are not the same, seek a way to rewrite them in terms of a common base.
02

Setting the Exponents Equal to Each Other

Once both sides of the equation are expressed as powers of the same base, it can be concluded that the exponents must be equal to each other for the entire equation to hold true. This stems from the fundamental rule of exponents stating that if a^m = a^n, then m must equal n.
03

Solve the Resulting Equation

With the exponents now set equal to each other, what remains is a simpler equation to solve. Standard methods of solving equations, like isolating variables and simplifying, can now be used. Finding the value of the variable in this step gives the answer of the original equation.

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Most popular questions from this chapter

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human ear. The formula $$D=10\left(\log I-\log I_{0}\right)$$ describes the loudness level of a sound, \(D,\) in decibels, where \(I\) is the intensity of the sound, in watts per meter \(^{2}\), and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question. If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

Approximate each number using a calculator. Round your answer to three decimal places. $$e^{-0.95}$$

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

The function $$N(t)=\frac{30,000}{1+20 e^{-1.5 t}}$$ describes the number of people, \(N(t),\) who become ill with influenza \(t\) weeks after its initial outbreak in a town with \(30,000\) inhabitants. The horizontal asymptote in the graph indicates that there is a limit to the epidemic's growth. (GRAPH CAN'T COPY) a. How many people became ill with the flu when the epidemic began? (When the epidemic began, \(t=0\).) b. How many people were ill by the end of the third week? c. Why can't the spread of an epidemic simply grow indefinitely? What does the horizontal asymptote shown in the graph indicate about the limiting size of the population that becomes ill?

I can solve \(4^{x}=15\) by writing the equation in logarithmic form.
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