Which of the following correctly solves the equation log(x−5)−log(−2x+13)=0

Question

log
⁡
(
𝑥
−
5
)
−
log
⁡
(
−
2
𝑥
+
13
)
=
0
?(1 point)
Responses

x=−6
𝑥
=
−
6
x equals negative 6

x=112
𝑥
=
11
2
x equals Start Fraction 11 over 2 End Fraction

x=6
𝑥
=
6
x equals 6

no solution

Answer 1

To solve the equation

\[ \log(x - 5) - \log(-2x + 13) = 0, \]

we can use the properties of logarithms. Recall that the difference of two logarithms can be combined into a single logarithm:

\[ \log \left( \frac{x - 5}{-2x + 13} \right) = 0. \]

This implies:

\[ \frac{x - 5}{-2x + 13} = 1. \]

Multiplying both sides by \(-2x + 13\) (assuming \(-2x + 13 \neq 0\)), we get:

\[ x - 5 = -2x + 13. \]

Now, adding \(2x\) to both sides results in:

\[ 3x - 5 = 13. \]

Next, we add \(5\) to both sides:

\[ 3x = 18. \]

Dividing both sides by \(3\), we find:

\[ x = 6. \]

Now, we need to check if \(x = 6\) results in valid arguments for the logarithm. Specifically, we check:

1. \(x - 5 = 6 - 5 = 1 > 0\) (valid argument)
2. \(-2x + 13 = -2(6) + 13 = -12 + 13 = 1 > 0\) (valid argument)

Both arguments are positive, so \(x = 6\) is a valid solution.

Hence, the correct answer is:

\[ \boxed{x = 6} \]