/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Solve each equation. $$ 3^{x... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 22

Solve each equation. $$ 3^{x}=\frac{1}{81} $$

Short Answer

Expert verified
x = -4

Step by step solution

01

Write the Equation in Exponential Form

First, express both sides of the equation as powers of the same base. Recognize that 81 is a power of 3, since 81 = 3^4. Therefore, \( \frac{1}{81} \) can be written as \( 3^{-4} \). So, rewrite the equation: \( 3^x = 3^{-4} \).
02

Set the Exponents Equal

Since the bases are the same (both are 3), the exponents must be equal. Therefore, we set the exponents equal to each other: \( x = -4 \).
03

Verify the Solution

Substitute \( x = -4 \) back into the original equation to verify. Thus, \( 3^{-4} = \frac{1}{81} \) is true, confirming the solution.

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Key Concepts

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

exponential form
In algebra, an **exponential equation** involves an unknown variable appearing as an exponent. To solve these equations, converting them to the same base is crucial.
Exponential form means expressing numbers as powers of a base number, typically a small prime number like 2, 3, or 5. For instance, 16 in exponential form is written as \(2^4\), since 2 multiplied by itself 4 times equals 16.
To solve the given problem \(3^x = \frac{1}{81}\), notice that 81 can be written as a power of 3: \(81 = 3^4\). Therefore, \(\frac{1}{81}\) can be expressed as \(3^{-4}\). Converting both sides to the same base helps simplify and solve the equation.
equating exponents
After rewriting both sides of the equation with the same base, we can equate the exponents directly. This works because the bases are identical. For example:
Given the rewritten equation \(3^x = 3^{-4}\),
both sides have the base 3. Therefore, the exponents must be equal:
  • Since \(3^x = 3^{-4}\),
  • We set the exponents equal: \(x = -4\)
This step significantly simplifies the problem. It turns an exponential equation into a basic linear equation, which is much easier to solve.
verification of solution
Verification ensures that our solution is correct. To check if \(x = -4\) is the right answer, substitute it back into the original equation.
Replace x with -4:
  • Left side: \(3^{-4}\)
  • Right side: \(\frac{1}{81}\)
Calculate the left side:
\(3^{-4} = \frac{1}{3^4} = \frac{1}{81}\)
Both sides are equal, confirming that \(x = -4\) is indeed the correct solution.
This final step is crucial, as it assures that our transformations and calculations are accurate.

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Most popular questions from this chapter

Let \(f(x)=2^{x} .\) We will see in the next section that this function is one- toone. Find each value, always working part (a) before part \((b)\). (a) \(f(4)\) (b) \(f^{-1}(16)\)

Find each logarithm. Give approximations to four decimal places. \(\ln 7.84\)

Simplify each expression. Write answers using only positive exponents. See Sections 4.1 and 4.2. $$ \frac{7^{8}}{7^{-4}} $$

Without using a calculator, give the value of \(\ln e^{\sqrt{3}}\).

Each function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window. \(f(x)=2 x-7\)
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