point(s) possible $$ \log (8 x+9)=1+\log (x-4) $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\log(8x + 9) = 1 + \log(x - 4)
$$

**Step 1: Use Logarithm Properties**

First, isolate the logarithmic terms:

$$
\log(8x + 9) - \log(x - 4) = 1
$$

Using the logarithm subtraction rule:

$$
\log\left(\frac{8x + 9}{x - 4}\right) = 1
$$

**Step 2: Exponentiate Both Sides**

To eliminate the logarithm, exponentiate both sides with base 10:

$$
\frac{8x + 9}{x - 4} = 10^1
$$

$$
\frac{8x + 9}{x - 4} = 10
$$

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

Multiply both sides by $$ x - 4 $$:

$$
8x + 9 = 10(x - 4)
$$

Expand and simplify:

$$
8x + 9 = 10x - 40
$$

$$
9 + 40 = 10x - 8x
$$

$$
49 = 2x
$$

$$
x = \frac{49}{2} = 24.5
$$

**Step 4: Verify the Solution**

Ensure that the solution lies within the domain of the original logarithmic expressions:

$$
x - 4 > 0 \implies x > 4 \
8x + 9 > 0 \implies x > -\frac{9}{8}
$$

Since $$ x = 24.5 $$ satisfies both conditions, it is a valid solution.

**Conclusion**

There is **one unique solution** to the equation:

$$
x = 24.5
$$
The solution to the equation $$ \log(8x + 9) = 1 + \log(x - 4) $$ is $$ x = 24.5 $$.

point(s) possible