Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₂(x+2)−log₂(x−5)=3

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Recall the logarithmic property that states \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \). Apply this to the equation \( \log_2 (x+2) - \log_2 (x-5) = 3 \) to combine the logs into one: \( \log_2 \left( \frac{x+2}{x-5} \right) = 3 \).

Rewrite the logarithmic equation in its equivalent exponential form. Since \( \log_b y = x \) means \( y = b^x \), rewrite \( \log_2 \left( \frac{x+2}{x-5} \right) = 3 \) as \( \frac{x+2}{x-5} = 2^3 \).

Simplify the right side of the equation: \( 2^3 = 8 \), so the equation becomes \( \frac{x+2}{x-5} = 8 \).

Solve the resulting rational equation by cross-multiplying: \( x + 2 = 8(x - 5) \). Then, distribute and collect like terms to isolate \( x \).

Check the domain restrictions for the original logarithmic expressions: \( x + 2 > 0 \) and \( x - 5 > 0 \), which means \( x > -2 \) and \( x > 5 \). Only values of \( x \) greater than 5 are valid. Verify that your solution satisfies this domain before finalizing your answer.

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Properties of Logarithms

Understanding the properties of logarithms, especially the subtraction rule log_b(A) - log_b(B) = log_b(A/B), is essential for simplifying and solving logarithmic equations. This allows combining multiple logarithmic terms into a single logarithm, making the equation easier to solve.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) includes only positive values of x. When solving logarithmic equations, it is crucial to identify and exclude any solutions that make the arguments of the logarithms non-positive, ensuring the solution is valid within the function's domain.

Converting Logarithmic Equations to Exponential Form

To solve logarithmic equations, converting the equation from logarithmic form to exponential form is often necessary. For example, log_b(A) = C can be rewritten as A = b^C, which allows solving for the variable algebraically.