We are given the equation:

$\log_3(x - 8) + \log_3(x - 3) - \log_3(x) = 1$

We can simplify this using the properties of logarithms.

Step 1: Combine the logarithmic terms

We can use the logarithmic property that states:

$\log_b(A) + \log_b(B) = \log_b(A \cdot B)$

Thus:

$\log_3((x - 8)(x - 3)) - \log_3(x) = 1$

Next, we use the logarithmic subtraction property:

$\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$

So, this becomes:

$\log_3\left(\frac{(x - 8)(x - 3)}{x}\right) = 1$

Step 2: Exponentiate both sides to eliminate the logarithm

Since $\log_3(A) = 1$ implies $A = 3^1$, we can rewrite the equation as:

$\frac{(x - 8)(x - 3)}{x} = 3$

Step 3: Solve the equation

Multiply both sides by $x$ to eliminate the denominator:

$(x - 8)(x - 3) = 3x$

Expand the left-hand side:

$x^2 - 3x - 8x + 24 = 3x$

Simplify:

$x^2 - 11x + 24 = 3x$

Move all terms to one side:

$x^2 - 14x + 24 = 0$

Step 4: Solve the quadratic equation

Now we solve the quadratic equation $x^2 - 14x + 24 = 0$. Using the quadratic formula:

$x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(24)}}{2(1)}$

Simplify:

$x = \frac{14 \pm \sqrt{196 - 96}}{2}$

$x = \frac{14 \pm \sqrt{100}}{2}$

$x = \frac{14 \pm 10}{2}$

So, we have two solutions:

$x = \frac{14 + 10}{2} = \frac{24}{2} = 12$

$x = \frac{14 - 10}{2} = \frac{4}{2} = 2$

Step 5: Check for extraneous solutions

We need to check if these solutions make the arguments inside the logarithms positive. The arguments must satisfy $x - 8 > 0$, $x - 3 > 0$, and $x > 0$.

For $x = 12$:

• $12 - 8 = 4 > 0$
• $12 - 3 = 9 > 0$
• $12 > 0$

So, $x = 12$ is a valid solution.

For $x = 2$:

• $2 - 8 = -6$, which is negative.

Thus, $x = 2$ is not a valid solution because it makes one of the logarithmic arguments negative.

Final Solution:

The only valid solution is $x = 12$.

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Would you like further details or have any questions?

Related Questions:

1. How can logarithmic properties be applied to different bases?
2. What are the steps to solve logarithmic inequalities?
3. How do you graph logarithmic functions and their transformations?
4. Can you explain how to solve logarithmic equations involving exponential terms?
5. What are extraneous solutions, and how can you check for them?

Tip: Always check for extraneous solutions in logarithmic equations to ensure that the solution doesn't produce undefined expressions.