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Chapter 12: Problem 38

Solve each equation. Give exact solutions. $$ \log _{6}\left(x^{2}+11\right)=2 $$

Short Answer

Expert verified
The solutions are \(x = 5\) and \(x = -5\).

Step by step solution

01

Rewrite the logarithmic equation in exponential form

The given equation is \(\log _{6}(x^{2}+11)=2\). To eliminate the logarithm, rewrite it in its exponential form. The base is 6, the exponent is 2, and it equals the argument of the log function. This gives us: \[6^2 = x^2 + 11\]
02

Simplify the exponential equation

Calculate \(6^2\): \[36 = x^2 + 11\]
03

Solve for \(x^2\)

Isolate \(x^2\) by subtracting 11 from both sides of the equation: \[36 - 11 = x^2\] \[25 = x^2\]
04

Solve for \(x\)

Take the square root of both sides to find the values of \(x\): \[x = \pm \sqrt{25}\] \[x = \pm 5\]

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Key Concepts

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

logarithmic functions
Logarithmic functions are incredibly important in mathematics because they help us solve equations involving exponential growth or decay. When you see an equation with a logarithm, like \(\text{log}_{6}(x^{2}+11)=2\), you're dealing with a logarithm to the base 6. The key idea is that logarithms and exponents are inverses of each other. This means you can rewrite the logarithmic equation in an exponential form, which often makes it easier to solve.
Logarithms have specific rules like:
  • Product Rule: \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n)\
  • Quotient Rule: \text{log}_b\frac{m}{n} = \text{log}_b(m) - \text{log}_b(n)\
  • Power Rule: \text{log}_b(m^n) = n \text{log}_b(m)\

For our problem, \(\text{log}_{6}(x^{2}+11) = 2\), converting it using the exponential form helps us to isolate and solve for x.
exponential form
Rewriting logarithmic equations in exponential form can make complex problems look much simpler. In the provided exercise, we used the exponential form to reframe the problem:
Given: \(\text{log}_{6}(x^{2}+11)=2\)
Exponential form: \[6^2 = x^2 + 11 \]
Here 6 is the base of the exponential equation, 2 is the exponent, and \(x^2 + 11 \) is the result.
This step effectively eliminates the logarithm, making the equation straightforward to solve.
One important thing to note here: Always remember the base of the logarithm becomes the base of the exponent in exponential form. And the integer equates to the exponent.
solving equations
Solving equations involves finding the values that satisfy the equation. Once we have rewritten the logarithmic equation in its exponential form, solving it becomes much easier. From the exercise:
\[6^2 = x^2 + 11\]
Calculate \({6^2 = 36}\), which simplifies the equation to:
\({36 = x^2 + 11}\).
Next, isolate \(x^2\) by subtracting 11 from both sides:
\

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Most popular questions from this chapter

Determine whether each statement is true or false. $$\log _{3} 7+\log _{3} 7^{-1}=0$$

Solve each equation. \(x=\log _{27} 3\)

Suppose that you are an agent for a detective agency. Today's encoding function is \(f(x)=4 x-5 .\) Find the rule for \(f^{-1}\) algebraically.

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{7}{5}$$

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