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Chapter 10: Problem 322

In the following exercises, solve each equation. \(3^{3 x+1}=81\)

Short Answer

Expert verified
x = 1

Step by step solution

01

Write 81 as a power of 3

Recognize that 81 can be written as a power of 3. Since 81 = 3^4, then we can rewrite the equation as: 3^{3x+1} = 3^4
02

Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal to each other: 3x + 1 = 4
03

Solve for x

Subtract 1 from both sides to isolate 3x:3x = 3Now, divide by 3:x = 1

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Key Concepts

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

Expressing numbers as powers
To solve exponential equations like the one given, recognizing numbers as powers of other numbers is crucial. For example, in the equation \(3^{3x+1} = 81\), we notice that 81 is a power of 3. Knowing that 81 can be expressed as \(3^4\) makes it easier to work with the equation. When trying to express a number as a power, consider the base numbers (like 2, 3, 4, etc.) and see if repeated multiplication of the base gives you the original number. Here are steps to help you with this:
  • Start with smaller bases like 2, 3, 4, or 5.
  • Keep multiplying the base by itself until you either get the original number or exceed it.
  • If the multiplication equates to the number, you鈥檝e found the exponent. Otherwise, try a different base.
Example: Recognize \(16\) as a power of 2. Since \(2^4 = 16\), we know \(16\) is \(2^4\). This process helps simplify exponential equations.
Setting exponents equal
Once you have expressed the numbers as powers with the same base, the next step is setting the exponents equal. In our example \(3^{3x+1} = 3^4\), since both sides of the equation have the same base (which is 3), you can just set their exponents equal: \(3x+1 = 4\). This works because if \(a^m = a^n\), then \(m = n\). This step simplifies your equation significantly. Key points to remember:
* Ensure both numbers are expressed with the same base.
* Carefully set the exponents equal to each other.
* Double-check your work to avoid mistakes.
Solving linear equations
After setting the exponents equal, the last step is solving the resulting linear equation. In this case, we have \(3x+1 = 4\). To solve for \(x\), follow these steps:
  • Subtract 1 from both sides of the equation to isolate the term with \(x\): \(3x+1-1 = 4-1\). This simplifies to \(3x = 3\).
  • Next, divide both sides by 3 to solve for \(x\): \(3x/3 = 3/3\). This simplifies to \(x = 1\).
These steps are part of what we call 'solving linear equations.' Always perform the same operation on both sides to maintain balance. Check if your result satisfies the original equation: If \(x = 1\), substitute \(x\) back into the exponent: \(3^{3(1)+1} = 3^4 = 81\). Your solution is verified!

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Most popular questions from this chapter

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm. \(\log _{12} 87\)

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In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log _{3} x^{2}\)

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