Solve each equation. Give solutions in exact form. See Examples 5–9. log(9x + 5) = 3 + log(x + 2)

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Start with the given equation: \(\log(9x + 5) = 3 + \log(x + 2)\).

Use the logarithm property to isolate the logarithmic terms on one side: subtract \(\log(x + 2)\) from both sides to get \(\log(9x + 5) - \log(x + 2) = 3\).

Apply the logarithmic subtraction rule: \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\), so rewrite the left side as \(\log\left(\frac{9x + 5}{x + 2}\right) = 3\).

Rewrite the equation in exponential form to eliminate the logarithm: \(\frac{9x + 5}{x + 2} = 10^{3}\), since the base of the logarithm is 10.

Solve the resulting equation \(\frac{9x + 5}{x + 2} = 1000\) by cross-multiplying and then isolating \(x\) to find the exact solution(s).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating and simplifying logarithmic expressions. In this problem, the ability to combine or separate logs helps isolate the variable and solve the equation.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation to isolate the logarithmic expressions, then converting the logarithmic form to exponential form. This process allows for solving for the variable inside the logarithm.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations involving logs, it is crucial to check that the solutions do not make any log argument zero or negative, ensuring the solutions are valid within the domain.

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