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−6)+log₂(x−4)−log₂ x=2

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Identify the domain restrictions for the logarithmic expressions: since the arguments of the logarithms must be positive, set up the inequalities \(x - 6 > 0\), \(x - 4 > 0\), and \(x > 0\). Solve these to find the domain of \(x\).

Use the logarithmic property that \(\log_b A + \log_b B = \log_b (A \times B)\) to combine the first two logarithms: \(\log_2 (x - 6) + \log_2 (x - 4) = \log_2 ((x - 6)(x - 4))\).

Apply the logarithmic property that \(\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)\) to combine all logarithms into a single logarithm: \(\log_2 \left( \frac{(x - 6)(x - 4)}{x} \right) = 2\).

Rewrite the equation \(\log_2 \left( \frac{(x - 6)(x - 4)}{x} \right) = 2\) in its exponential form: \(\frac{(x - 6)(x - 4)}{x} = 2^2\).

Solve the resulting algebraic equation for \(x\), then check each solution against the domain restrictions found in step 1 to reject any extraneous solutions.

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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 combining or simplifying logarithmic expressions. For example, log_b(A) + log_b(B) = log_b(AB) and log_b(A) - log_b(B) = log_b(A/B). These rules allow the equation to be rewritten in a simpler form to solve for x.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive (x > 0). When solving logarithmic equations, it is crucial to check that the solutions do not make any logarithm's argument zero or negative, as these values are not valid and must be rejected.

Solving Exponential Equations

After applying logarithmic properties, the equation often converts into an exponential form. Solving the resulting polynomial or exponential equation involves algebraic manipulation to find exact solutions. These solutions can then be approximated using a calculator if needed.

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