Solve for X:
(x-6)log4=(3x)log2
(x-6)*Log4 = 3x*Log2.
Divide by Log4:
x-6 = 3x*Log2/Log4.
x-6 = 3x*0.5 = 1.5x.
x-1.5x = 6.
-0.5x = 6.
X = -12.
To solve for X in the equation:
(x - 6) log4 = (3x) log2,
we can use the properties of logarithms to simplify the equation and isolate X.
First, let's use the log property:
loga - logb = log(a / b).
Applying this property, we can rewrite the equation as:
log4 [(x - 6)] = log2 [(3x)] / log2 [4].
Since log2 [4] is equal to 2 (because 2 raised to the power of 2 gives 4), we simplify further:
log4 [(x - 6)] = (log2 [(3x)]) / 2.
Next, we can apply the property:
loga [bx] = x loga [b].
Using this property, we get:
(x - 6) = (log2 [(3x)]) / 2 log4.
To solve for X, we need to isolate it on one side of the equation. Let's multiply both sides of the equation by 2 log4:
2 log4 [(x - 6)] = log2 [(3x)].
Now, we can apply the property:
loga [b^c] = c loga [b].
Using this property, we simplify further:
log4 [(x - 6)^2] = log2 [(3x)].
We can now equate the expressions within the logarithms:
(x - 6)^2 = 3x.
Expanding the left side of the equation:
x^2 - 12x + 36 = 3x.
Combining like terms and bringing all terms to one side of the equation:
x^2 - 15x + 36 = 0.
Finally, we have a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula. Solving this quadratic equation would give us the values of X that satisfy the original equation.