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Chapter 9: Problem 9

Solve. Where appropriate, include approximations to three decimal places. $$ 3^{2 x}=81 $$

Short Answer

Expert verified
x = 2

Step by step solution

01

Understand the given equation

The equation given is \[ 3^{2x} = 81 \].Identify the base and exponent on each side of the equation.
02

Express 81 as a power of 3

Note that 81 can be expressed as a power of 3. Since \[ 81 = 3^4 \].We can rewrite the equation \[ 3^{2x} = 3^4 \].
03

Set the exponents equal

Since the bases are the same (both are 3), we can set the exponents equal to each other:\[ 2x = 4 \].
04

Solve for x

Solve the equation \[ 2x = 4 \]by dividing both sides by 2:\[ x = \frac{4}{2} = 2 \].
05

Approximation (if needed)

In this particular problem, the answer is an integer. No approximation is needed. If necessary, the answer could be approximated to three decimal places as \[ x = 2.000 \].

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number being multiplied, while the exponent tells us how many times to multiply the base by itself. For example, in the expression \(3^4\), 3 is the base, and 4 is the exponent, resulting in \(3 \times 3 \times 3 \times 3 = 81\). Understanding how to manipulate and simplify exponents is crucial for solving exponential equations.

Intermediate steps such as expressing numbers as powers of a smaller base, like writing 81 as \(3^4\), can simplify complex problems. This knowledge helps us recognize patterns and solve equations efficiently.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. For exponential equations like \(3^{2x} = 81\), we first express all terms with a common base. By rewriting 81 as \(3^4\), we can transform the original equation into \(3^{2x} = 3^4\).

With the bases identical on both sides, we can then focus on the exponents alone, setting them equal to each other: \(2x = 4\). Solving for \(x\) by dividing both sides by 2, we get \(x = 2\). This step-by-step approach breaks down the problem into manageable parts.
Powers of Numbers
Powers of numbers represent repeated multiplication of a base number. Understanding how to work with powers is essential for tackling many algebraic problems. For instance, recognizing that \(81 = 3^4\) allows us to equate powers with the same base when solving equations.

This concept helps in simplifying and solving exponential equations. By rewriting both sides of the equation with the same base, we can subtract the complexity and focus on the exponents, making it easier to find the solution.

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Most popular questions from this chapter

Simplify. \(3^{3}+2^{2}-(32 \div 4-16 \div 8)\)

Simplify. $$ \log _{81} 3 \cdot \log _{3} 81 $$

The average price paid for a Super Bowl ticket has increased exponentially from 12 dollars in 1967 to 2670 dollars in 2015 Data: DallasNews.com; seatgeek.com a) Find the exponential growth rate \(k,\) and write an equation for an exponential function that can be used to predict the average price paid for a Super Bowl ticket \(t\) years after 1967 b) Estimate the year in which the average price paid for a Super Bowl ticket will reach 5000 dollars.

Graph both equations using one set of axes: $$ y=\left(\frac{3}{2}\right)^{x}, \quad y=\log _{3 / 2} x $$

Solve. If no solution exists, state this. $$ 3^{x^{2}} \cdot 3^{4 x}=\frac{1}{27} $$
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