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Chapter 10: Problem 37

Solve each equation. Give exact solutions. $$ \log _{2}\left(x^{2}+7\right)=4 $$

Short Answer

Expert verified
The exact solutions are \(x = 3\) and \(x = -3\).

Step by step solution

01

Understand the given logarithmic equation

The given equation is \ \ \ \ \ \ \( \log _{2}(x^{2}+7)=4 \). This means that the logarithm of \(x^{2}+7\) to the base 2 equals 4.
02

Convert the logarithmic equation to an exponential equation

Use the property of logarithms: \( \log _{b}(a)=c \rightarrow a=b^{c} \). So, \( x^{2}+7 = 2^{4} \).
03

Simplify the exponential equation

Calculate \(2^{4}\): \(2^{4} = 16\). Now the equation becomes \( x^{2} + 7 = 16 \).
04

Solve for \(x^{2}\)

Subtract 7 from both sides of the equation: \( x^{2} = 16 - 7 \rightarrow x^{2} = 9 \).
05

Solve for \(x\)

Take the square root of both sides: \( x = \pm \sqrt{9} \rightarrow x = \pm 3 \).

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Key Concepts

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

Logarithmic Functions
Let's start by understanding logarithmic functions. A logarithmic function is the inverse of an exponential function. If you have a logarithmic function written as \( \log _{b}(a) = c \), it means that \( b^c = a \). In our exercise, \( \log _{2}(x^{2}+7) = 4 \) means we are looking for a value that, when 2 is raised to this power, it equals \( x^{2} + 7 \). Breaking down logarithms is essential to solving these types of equations.
Exponential Equations
Being familiar with exponential equations is crucial here. Exponential equations are equations in which variables appear as exponents. For example, in our problem, we converted the logarithmic equation \( \log _{2}(x^{2}+7) = 4 \) to an exponential form: \( x^{2}+7 = 2^{4} \). Essentially, this means we are finding a number which, when 2 is raised to the 4th power, equals that number. Understanding this conversion is an essential step.
Solving for x
Once we have our exponential equation, solving for \( x \) involves basic algebra. First, we calculate \( 2^4 = 16 \). So, we rewrite the equation as \( x^{2} + 7 = 16 \). Next, we isolate \( x^2 \) by subtracting 7: \( x^{2} = 9 \). Finally, we solve for \( x \) by taking the square root: \( x = \pm \sqrt{9} \). Therefore, \( x = 3 \) or \( x = -3 \). This is how we find the solutions for \( x \) in a simple logarithmic equation. Always remember to check your solutions in the original equation to verify that they work!

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Most popular questions from this chapter

Solve each equation. Approximate solutions to three decimal places. $$ 7^{x}=5 $$

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Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{\ln 2 x}=e^{\ln (x+1)} $$

Graph each logarithmic function. $$f(x)=\log _{4} x$$

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