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

Verified step by step guidance

1. Recall the logarithmic property that allows combining sums and differences of logs with the same base: $\log 2_{}(a)\; +\; log{2}_{}(b)\; =\; log{2}_{}(ab)$ and $log$2(a) - log2(b) = log2(a/b).
2. Apply these properties to the equation $log\_2(x-6)\; +\; log\_2(x-4)\; -\; log\_2\; x\; =\; 2$ to combine the logarithms into a single log expression: $log\_2\; \backslash left(\; \backslash frac\{(x-6)(x-4)\}\{x\}\; \backslash right)\; =\; 2$.
3. Rewrite the logarithmic equation in its equivalent exponential form: $\backslash frac\{(x-6)(x-4)\}\{x\}\; =\; 2^2$. Since $2^2\; =\; 4$, this becomes $\backslash frac\{(x-6)(x-4)\}\{x\}\; =\; 4$.
4. Multiply both sides of the equation by $x$ to clear the denominator: $(x-6)(x-4)\; =\; 4x$. Then expand the left side using the distributive property: $x^2\; -\; 4x\; -\; 6x\; +\; 24\; =\; 4x$.
5. Simplify and rearrange the equation to standard quadratic form: $x^2\; -\; 10x\; +\; 24\; =\; 4x$ becomes $x^2\; -\; 14x\; +\; 24\; =\; 0$. This quadratic can then be solved using factoring, completing the square, or the quadratic formula.

Key Concepts

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 for easier solving.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all input values for which the argument is positive. Since log_b(x) is only defined for x > 0, any solution must satisfy the conditions x−6 > 0, x−4 > 0, and x > 0. Checking the domain ensures that no extraneous solutions are accepted.

Solving Exponential Equations

After applying logarithmic properties, the equation often converts to an exponential form. Solving exponential equations involves isolating the variable and using inverse operations. For example, if log_b(A) = C, then A = b^C. This step is crucial to find the exact value of x before approximation.