Felect the correct next step. Find the solution. $$ \begin{array}{l}\log _{4}\left(\frac{8 x}{x-7}\right)=3 \ \frac{8 x}{x-7}=4^{3} \quad \frac{8 x}{x-7}=3 \cdot 4 \quad \frac{8 x}{x-7}=3^{4} \ \end{array} $$

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UpStudy AI · Accepted Answer

To solve the equation:

$$
\log _{4}\left(\frac{8 x}{x-7}\right)=3
$$

**Step 1: Apply the Definition of a Logarithm**

The equation $$\log_b(a) = c$$ is equivalent to $$a = b^c$$. Applying this definition:

$$
\frac{8x}{x - 7} = 4^3
$$

**Step 2: Calculate $$4^3$$**

$$
4^3 = 64
$$

So, the equation becomes:

$$
\frac{8x}{x - 7} = 64
$$

**Step 3: Solve for $$x$$**

Multiply both sides by $$x - 7$$ to eliminate the denominator:

$$
8x = 64(x - 7)
$$

Expand the right side:

$$
8x = 64x - 448
$$

Subtract $$64x$$ from both sides:

$$
-56x = -448
$$

Divide both sides by $$-56$$:

$$
x = 8
$$

**Step 4: Verify the Solution**

Ensure that the solution does not make the argument of the logarithm non-positive:

$$
x - 7 = 8 - 7 = 1 > 0 \quad \text{and} \quad \frac{8 \times 8}{1} = 64 > 0
$$

Both conditions are satisfied.

**Final Answer:**

$$\boxed{8}$$
The solution is $$ x = 8 $$.

Felect the correct next step. Find the solution. \[ \begin{array}{l}\log _{4}\left(\frac{8 x}{x-7}\right)=3 \\ \frac{8 x}{x-7}=4^{3} \quad \frac{8 x}{x-7}=3 \cdot 4 \quad \frac{8 x}{x-7}=3^{4} \\ \end{array} \]

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