log3 (2x – 1) – log3 (x – 4) = 2
Que significa^ese signo
0949510891
To solve the given logarithmic equation log3 (2x – 1) – log3 (x – 4) = 2, we can use the properties of logarithms to simplify and manipulate the equation.
Step 1: Apply the quotient rule of logarithms.
According to the quotient rule, loga(b) – loga(c) = loga(b/c).
Using this rule, we can rewrite the given equation as:
log3((2x – 1)/(x – 4)) = 2.
Step 2: Simplify the equation further.
Since the base of the logarithm is 3, if we raise 3 to the power of both sides, the equation becomes:
3^log3((2x – 1)/(x – 4)) = 3^2.
Step 3: Simplify and remove the logarithm.
Applying the property that a logarithm and its base inverse operation (exponentiation) cancel each other out, we get:
((2x – 1)/(x – 4)) = 9.
Step 4: Solve the equation.
Next, we can solve the equation for x by cross-multiplying:
9(x – 4) = 2x – 1.
Expanding and collecting like terms:
9x – 36 = 2x – 1.
Step 5: Further solve for x.
Move all the terms with x to one side of the equation and all the constants to the other side:
9x – 2x = 36 – 1,
7x = 35.
Finally, divide both sides of the equation by 7 to solve for x:
x = 35/7,
x = 5.
Therefore, the value of x that satisfies the original logarithmic equation log3 (2x – 1) – log3 (x – 4) = 2 is x = 5.
log3 (2x – 1) – log3 (x – 4) = 2 , clearly x > 4
(2x-1)/(x-4) = 3^2
2x^2 -9x-5 = 0
(2x+1)(x-5) = 0
x = -1/2 or x = 5
but x > 4, so
x = 5