For what value of $$ x $$ is the following true? $$ \log (x+8)=\log x+\log 8 $$ Your answer is: $$ x=\square $$ You may enter the exact value or round to 4 decimal places.

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To solve the equation

$$
\log (x+8)=\log x + \log 8,
$$

we first use the logarithm product rule:

$$
\log ab = \log a + \log b.
$$

Thus, the right-hand side becomes:

$$
\log x + \log 8 = \log (8x).
$$

The equation is now:

$$
\log (x+8)=\log (8x).
$$

Since the logarithm function is one-to-one, we equate the arguments:

$$
x+8 = 8x.
$$

Subtract $$ x $$ from both sides:

$$
8 = 7x.
$$

Solving for $$ x $$:

$$
x = \frac{8}{7}.
$$

To ensure the solution is valid, we verify the domain. In logarithmic functions, the arguments must be positive:
- For $$ \log x $$: $$ x > 0 $$.
- For $$ \log (x+8) $$: $$ x+8 > 0 $$ which simplifies to $$ x > -8 $$.

Since $$ \frac{8}{7} > 0 $$, the domain requirements are satisfied.

Thus, the answer is:

$$
x = \frac{8}{7}.
$$
$$ x = \frac{8}{7} $$

For what value of \( x \) is the following true? \( \log (x+8)=\log x+\log 8 \) Your answer is: \( x=\square \) You may enter the exact value or round to 4 decimal places.

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