To solve this logarithmic equation, we can use the properties of logarithms to simplify it.
First, we can combine the two logarithms on the left side using the product rule for logarithms: log2(4-x)(1-2x) = 2log2 3
Next, we can simplify the expression inside the logarithm on the left side: log2(4-x)(1-2x) = log2 3^2
Now, we can apply the power rule for logarithms to simplify further: (4-x)(1-2x) = 3^2
Expanding the left side of the equation: 4 - 4x - x + 2x^2 = 9
Rearranging the terms and simplifying: 2x^2 - 5x - 5 = 0
Now, we have a quadratic equation that we can solve using the quadratic formula: x = [5 ± sqrt(5^2 - 4(2)(-5))] / 2(2) x = [5 ± sqrt(25 + 40)] / 4 x = [5 ± sqrt(65)] / 4
Therefore, the solutions to the equation are: x = (5 + sqrt(65)) / 4 x = (5 - sqrt(65)) / 4
These are the values of x that satisfy the original logarithmic equation.
To solve this logarithmic equation, we can use the properties of logarithms to simplify it.
First, we can combine the two logarithms on the left side using the product rule for logarithms:
log2(4-x)(1-2x) = 2log2 3
Next, we can simplify the expression inside the logarithm on the left side:
log2(4-x)(1-2x) = log2 3^2
Now, we can apply the power rule for logarithms to simplify further:
(4-x)(1-2x) = 3^2
Expanding the left side of the equation:
4 - 4x - x + 2x^2 = 9
Rearranging the terms and simplifying:
2x^2 - 5x - 5 = 0
Now, we have a quadratic equation that we can solve using the quadratic formula:
x = [5 ± sqrt(5^2 - 4(2)(-5))] / 2(2)
x = [5 ± sqrt(25 + 40)] / 4
x = [5 ± sqrt(65)] / 4
Therefore, the solutions to the equation are:
x = (5 + sqrt(65)) / 4
x = (5 - sqrt(65)) / 4
These are the values of x that satisfy the original logarithmic equation.