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Chapter 3: Problem 9

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$32^{x}=8$$

Short Answer

Expert verified
The solution to the equation \(32^{x} = 8\) is \(x = 0.6\).

Step by step solution

01

Express the Numbers as Powers of the Same Base

Given the equation \(32^{x} = 8\), both 8 and 32 can be expressed as powers of 2. For instance 8 is \(2^{3}\), and 32 is \(2^{5}\). So, the given equation can be rewritten as \((2^{5})^{x} = 2^{3}\).
02

Simplify the Equation

In the equation \((2^{5})^{x} = 2^{3}\), the laws of exponents are used. An exponent of an exponent means you multiply the exponents. So the equation simplifies to \(2^{5x} = 2^{3}\).
03

Equating the Exponents

Now that the bases are the same, the exponents can be equated to each other to solve for the variable \(x\). Thus, \(5x = 3\).
04

Solve for the Variable

To solve for \(x\), the equation \(5x = 3\) is solved by dividing both sides by 5. This gives \(x = 3/5 = 0.6\) as the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laws of Exponents
The laws of exponents make it easier to work with exponential expressions. They help simplify calculations and solve complex equations. There are several key rules to keep in mind:
  • Product of Powers Rule: When multiplying like bases, you add their exponents. For example, \(a^m \times a^n = a^{m+n}\).

  • Power of a Power Rule: When taking an exponent to another exponent, you multiply the exponents. This is shown as \((a^m)^n = a^{mn}\).

  • Quotient of Powers Rule: When dividing like bases, you subtract the exponent of the denominator from the exponent of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).
Understanding these rules is essential when solving exponential equations. They allow you to manipulate exponents and reach solutions more efficiently.
Equating Exponents
Equating exponents is a technique used to solve equations where each side is expressed as a power of the same base. This is a powerful method for finding unknowns in exponential equations.
To use this method effectively:
  • Ensure both sides of the equation have the same base.

  • Once the bases are identical, set the exponents equal to one another. For instance, if \(a^b = a^c\), then \(b = c\).
Equating exponents is straightforward, and it simplifies finding solutions when exponential terms are involved. It eliminates complex calculations by reducing the problem to solving a simple linear equation. Keep this in mind when tackling similar exponential problems.
Expressing Numbers as Powers of the Same Base
When solving exponential equations, expressing numbers as powers of the same base is a crucial step. This approach involves breaking down numbers into their base-exponent forms.
  • Start by finding a common base. Often, numbers can be expressed as powers of a smaller number, such as 2, 3, or 5 depending on the numbers in the equation.

  • Convert each number to this common base. For example, in the equation \(32^x = 8\), both can be written with the base of 2, as \(32 = 2^5\) and \(8 = 2^3\).
Once converted, apply the laws of exponents to simplify the equation. Using a common base makes it easier to work with exponential equations and offers a clear path to solving them.

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Most popular questions from this chapter

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. The figure indicates that lemon juice has a pH of 2.3. What is the hydrogen ion concentration? b. Stomach acid has a pH that ranges from 1 to 3. What is the hydrogen ion concentration of the most acidic stomach? c. How many times greater is the hydrogen ion concentration of the acidic stomach in part (b) than the lemon juice in part (a)?

Graph: \(f(x)=\frac{4 x^{2}}{x^{2}-9}\) (Section \(2.6,\) Example 6 )

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(y=2+\log _{3} x, y=\log _{3}(x+2),\) and \(y=-\log _{3} x\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

Given \(f(x)=\frac{2}{x+1}\) and \(g(x)=\frac{1}{x},\) find each of the following: a. \((f \circ g)(x)\) b. the domain of \(f \circ g .\) (Section \(1.7,\) Example 6 )
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