We are given the equation:

$\log_2(x - 7) = 2 + \log_2(x - 4)$

We need to solve for $x$. Let's break it down step by step.

Step 1: Use logarithmic properties

Since we have logarithms on both sides, we can isolate them. First, subtract $\log_2(x - 4)$ from both sides:

$\log_2(x - 7) - \log_2(x - 4) = 2$

Step 2: Apply the logarithmic subtraction rule

The subtraction rule for logarithms states:

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

So, applying this rule:

$\log_2\left(\frac{x - 7}{x - 4}\right) = 2$

Step 3: Convert the logarithmic equation to exponential form

Now, rewrite the equation in its exponential form. The logarithmic equation $\log_b(A) = C$ is equivalent to $A = b^C$. So:

$\frac{x - 7}{x - 4} = 2^2$

This simplifies to:

$\frac{x - 7}{x - 4} = 4$

Step 4: Solve for $x$

Now, multiply both sides by $(x - 4)$ to eliminate the fraction:

$x - 7 = 4(x - 4)$

Expand the right-hand side:

$x - 7 = 4x - 16$

Now, solve for $x$ by bringing all $x$-terms to one side:

$x - 4x = -16 + 7$

This simplifies to:

$-3x = -9$

Divide both sides by $-3$:

$x = 3$

Step 5: Check the solution

We must check if $x = 3$ satisfies the original logarithmic conditions. Since the arguments inside the logarithms must be positive:

• $x - 7 = 3 - 7 = -4$ (which is negative)

Since $\log_2(x - 7)$ is undefined for negative numbers, $x = 3$ is not a valid solution.

Final Answer:

There is no solution to this equation.

Would you like further clarification or details? Here are 5 related questions to explore:

1. How do you convert logarithmic equations to exponential form?
2. What are the key logarithmic properties used in solving such equations?
3. How can you determine the domain of a logarithmic function?
4. Why are negative numbers excluded from the arguments of logarithms?
5. Can you apply logarithmic transformations to both sides of any equation?

Tip: Always check for restrictions in the domain when solving logarithmic equations.