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Chapter 10: Problem 297

In the following exercises, solve for \(x\). \(\log _{3} x+\log _{3}(x+6)=3\)

Short Answer

Expert verified
x = 3

Step by step solution

01

Apply the Product Rule of Logarithms

Use the product rule of logarithms \[ \log_a b + \log_a c = \log_a (bc) \] to combine the logarithms on the left side of the equation. \[ \log_3 x + \log_3 (x+6) = \log_3 [x(x+6)] = \log_3 (x^2 + 6x) \] The equation now becomes: \[ \log_3 (x^2 + 6x) = 3 \]
02

Remove the Logarithm

To eliminate the logarithm, rewrite the equation in exponential form. \[ \log_3 (x^2 + 6x) = 3 \implies x^2 + 6x = 3^3 \] Simplifying the exponential expression gives: \[ x^2 + 6x = 27 \]
03

Set Up a Quadratic Equation

Rearrange the equation to set it equal to zero. \[ x^2 + 6x - 27 = 0 \]
04

Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \[ x^2 + 6x - 27 = 0 \, a = 1, b = 6, \c = -27. \] Plug these values into the quadratic formula. \[ x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-27)}}{2(1)} = \frac{-6 \pm \sqrt{36 + 108}}{2} = \frac{-6 \pm \sqrt{144}}{2} = \frac{-6 \pm 12}{2} \] This results in two solutions: \[ x = \frac{-6 + 12}{2} = 3 \] and \[ x = \frac{-6 - 12}{2} = -9 \]
05

Verify the Solutions

Only consider solutions that make the original logarithmic statement true. Since the logarithm of a negative number is undefined, discard \ x = -9 \. Verify that \ x = 3 \ satisfies the original equation. \[ \log_3 3 + \log_3 (3+6) = 3 \] This simplifies to \[ \ 1 + 2 = 3 \]

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Key Concepts

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

logarithms
Logarithms are a way to express exponents and are the inverse operations of exponentiation. They tell us how many times one number must be multiplied by itself to produce another number. The general form of a logarithm is \ \ \ \[\ \ \log_b a = c\ \ \] \ \Here, \ \

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Most popular questions from this chapter

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{6} 4+\log _{6} 9\)

Use an example to show that $$ \log (a+b) \neq \log a+\log b $$

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{5} 2-\log _{5} x-\log _{5} y\)

In the following exercises, solve each equation. \(\log _{6} x+\log _{6}(x-5)=\log _{6} 24\)

In the following exercises, solve for \(x\), giving an exact answer as well as an approximation to three decimal places. \(6^{x}=91\)
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