Question

solve for (x).

\log_{2}(x-8) = 2 - \log_{2}(x-5)

if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

Explanation:

Determine the domain of the equation

To ensure the logarithms are defined, the arguments must be strictly positive:

$$ x - 8 > 0 \implies x > 8 $$

$$ x - 5 > 0 \implies x > 5 $$

Thus, the domain is \(x > 8\).

Apply properties of logarithms

Move the logarithmic terms to one side:

$$ \log_2(x - 8) + \log_2(x - 5) = 2 $$

Using the product rule for logarithms, \(\log_b(u) + \log_b(v) = \log_b(u \cdot v)\):

$$ \log_2((x - 8)(x - 5)) = 2 $$

Convert to exponential form

Rewrite the logarithmic equation in exponential form:

$$ (x - 8)(x - 5) = 2^2 $$

$$ (x - 8)(x - 5) = 4 $$

Solve the quadratic equation

Expand and simplify the equation:

$$ x^2 - 13x + 40 = 4 $$

$$ x^2 - 13x + 36 = 0 $$

Factor the quadratic equation:

$$ (x - 9)(x - 4) = 0 $$

This gives two potential solutions:

$$ x = 9 \quad \text{or} \quad x = 4 $$

Check for extraneous solutions

We must verify each solution against our domain restriction \(x > 8\):

• For \(x = 9\): Since \(9 > 8\), this is a valid solution.
• For \(x = 4\): Since \(4 \le 8\), this is an extraneous solution because \(\log_2(4-8) = \log_2(-4)\) is undefined.

Thus, the only valid solution is \(x = 9\).

Answer:

Solve for \(x\).

$$\log_{2}(x-8)=2-\log_{2}(x-5)$$

If there is more than one solution, separate them with commas.
If there is no solution, click on "No solution".

\(x =\) <blank>9</blank>