To solve the equation
[tex]\[ \log_4(2-x) = \log_4(-5x-18), \][/tex]
we will proceed with the following steps:
1. Recognize that if the logarithms are equal, then their arguments must be equal:
Since the bases of the logarithms on both sides of the equation are the same, we can equate their arguments:
[tex]\[ 2-x = -5x-18. \][/tex]
2. Solve the resulting linear equation:
We need to isolate \( x \). To do this, let's first add \( 5x \) to both sides of the equation to eliminate \( x \) on the right-hand side:
[tex]\[ 2 - x + 5x = -18. \][/tex]
Simplifying the left-hand side, we get:
[tex]\[ 2 + 4x = -18. \][/tex]
Next, subtract 2 from both sides to isolate the terms containing \( x \):
[tex]\[ 4x = -18 - 2, \][/tex]
which simplifies to:
[tex]\[ 4x = -20. \][/tex]
Now, divide both sides by 4:
[tex]\[ x = \frac{-20}{4}, \][/tex]
yielding:
[tex]\[ x = -5. \][/tex]
3. Verify the solution:
To ensure \( x = -5 \) is valid, substitute it back into the original arguments of the logarithms:
[tex]\[ 2 - (-5) = 2 + 5 = 7, \][/tex]
and
[tex]\[ -5(-5) - 18 = 25 - 18 = 7. \][/tex]
Since both arguments are indeed equal and positive (provided they must be positive to be in the domain of the logarithm for real numbers), \( x = -5 \) is a valid solution.
Thus, the solution to the equation is:
[tex]\[ x = -5. \][/tex]
