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Chapter 11: Problem 4

For Problems \(1-34\), solve each equation. $$ 3^{2 x}=81 $$

Short Answer

Expert verified
The solution is \(x = 2\).

Step by step solution

01

Identify Base Relationships

Recognize that both sides of the equation have powers of the base 3. We know that 81 can be expressed as a power of 3. Specifically, we have \(81 = 3^4\).
02

Rewrite the Equation

Substitute 81 with \(3^4\) in the equation to have the same base: \(3^{2x} = 3^4\).
03

Equate the Exponents

Since the bases are the same, you can set the exponents equal to each other. Hence, solve the equation \(2x = 4\).
04

Solve for x

Divide both sides of the equation by 2 to isolate \(x\): \[x = \frac{4}{2} = 2\].

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Key Concepts

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

Powers and Exponents
Exponents represent how many times a number, known as the base, is multiplied by itself. In the expression \(3^4\), 3 is the base and 4 is the exponent. This means that 3 is multiplied by itself four times: \(3 \times 3 \times 3 \times 3\). Similarly, in \(3^{2x}\), the base is 3, and the exponent is \(2x\), indicating that 3 is multiplied by itself \(2x\) times.

Understanding exponents is crucial when it comes to simplifying expressions and solving exponential equations. Here are a few key ideas about exponents to remember:
  • An exponent of 1 means the base's value remains unchanged, such as \(3^1 = 3\).
  • An exponent of 0 means the total value is 1, no matter the base: \(3^0 = 1\).
  • Negative exponents indicate a reciprocal: \(3^{-1} = \frac{1}{3}\).
Recognizing these rules can simplify solving equations involving exponents.
Equation Solving
Solving equations is about finding the value of the unknown that makes the equation true. When faced with exponential equations like \(3^{2x} = 81\), start by simplifying the equation so that both sides have the same base.

In our example, by rewriting 81 as \(3^4\), each side shares the base of 3, making it simpler to focus on the exponents. Once the bases are the same, the next step is to equate the exponents: this is because if \(a^m = a^n\), then it must be that \(m = n\). Therefore, you would solve \(2x = 4\).

Knowing how to manipulate equations is a fundamental skill. By maintaining balance and applying logical steps, you can confidently determine the unknown variable.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to find solutions or simplify expressions. Once you've equated exponents in our exercise, you handle the linear equation \(2x = 4\). This involves traditional algebraic methods to isolate and solve for \(x\).

To isolate \(x\), divide both sides by 2, resulting in: \[x = \frac{4}{2} = 2\]This method demonstrates how scaling back from exponential forms to simpler arithmetic equations can help in finding solutions efficiently.

Algebraic manipulation allows you to:
  • Simplify complex problems into manageable ones.
  • Reorganize equations to focus on essential variables.
  • Ensure precision through systematic logical steps.
By practicing these methods, you gain skills to tackle a range of mathematical challenges.

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Most popular questions from this chapter

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=-0.1452 $$

For Problems \(21-30\), use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 18\)

Graph \(f(x)=\log _{3} x\) by reflecting the graph of \(g(x)=3^{x}\) across the line \(y=x\).

Explain the concept of a Richter number.

For Problems \(1-14\), solve each exponential equation and express solutions to the nearest hundredth. $$ 3^{x-2}=11 $$
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