Solve for x:
2log(3-x) = log2 + log(22-2x)
2log(3-x) = log2 + log(22-2x)
log(3-x)^2 - log(22-2x) = log2
log[(3-x)^2/(22-2x)] = log2
(3-x)^2/(22-2x) = 2
(3-x)^2 = 2(22-2x)
expand, express and solve as a quadratic, make sure x < 3
To solve the equation `2log(3-x) = log2 + log(22-2x)` for `x`, we need to apply the properties of logarithms.
First, let's simplify the equation:
Using the property log(a) + log(b) = log(a * b), we can rewrite the equation as:
2log(3-x) = log(2 * (22-2x))
Next, applying the property log(a^b) = b * log(a), we can simplify further:
2log(3-x) = log(44 - 4x)
Now, let's convert the base of the logarithms to be the same. We can use the natural logarithm (ln) as the base.
Using the property log(base a) x = log(base b) x / log(base b) a, we have:
2ln(3-x) = ln(44 - 4x) / ln(10)
Next, we can apply the property ln(a * b) = ln(a) + ln(b) to simplify the right side of the equation:
2ln(3-x) = (ln(44) + ln(4x)) / ln(10)
Now, let's simplify the equation further by removing the denominator:
2ln(3-x) * ln(10) = ln(44) + ln(4x)
Using the property ln(a^b) = b * ln(a), we can rewrite the equation as:
ln((3-x)^2) * ln(10) = ln(44) + ln(4x)
Next, applying the property ln(a + b) = ln(a) + ln(b), we can simplify the right side of the equation:
ln((3-x)^2) * ln(10) = ln(44 * 4x)
Now, we can eliminate the logarithms by exponentiating both sides of the equation:
e^(ln((3-x)^2) * ln(10)) = e^(ln(44 * 4x))
Simplifying:
(3-x)^2 * 10 = 44 * 4x
Expanding the square:
9 - 6x + x^2 * 10 = 176x
Rearranging the equation:
10x^2 - 182x + 9 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (182 ± sqrt(182^2 - 4*10*9)) / (2*10)
Calculating:
x ≈ 1.0505 or x ≈ 0.0854
Therefore, the possible solutions for x are approximately 1.0505 or 0.0854.
To solve for x in the equation 2log(3-x) = log2 + log(22-2x), we will use logarithmic properties and algebraic manipulation steps. Here's how to approach it:
Step 1: Combine the logs
Using the logarithmic property log(a) + log(b) = log(a * b), rewrite the equation as:
log((3-x)²) = log(2 * (22-2x))
Step 2: Convert to exponential form
Apply the definition of the logarithm: if log(a) = log(b), then a = b. Rewrite the equation in exponential form:
(3-x)² = 2 * (22-2x)
Step 3: Expand and simplify
Expand both sides of the equation:
9 - 6x + x² = 44 - 4x
Step 4: Rearrange and combine like terms
Rearrange the equation into standard quadratic form:
x² - 2x - 35 = 0
Step 5: Solve the quadratic equation
Factor or use the quadratic formula to solve for x. In this case, the quadratic factors as: (x - 7)(x + 5) = 0
So, either x - 7 = 0 (which gives x = 7) or x + 5 = 0 (which gives x = -5)
Hence, the solution to the equation 2log(3-x) = log2 + log(22-2x) is x = 7 or x = -5.