Solve for x in the equation:log(5x-4)=log(x+2)+1/3
To solve for x, first simplify the equation:
log(5x-4) = log(x+2) + 1/3
Using the logarithmic property that log(a) + log(b) = log(ab), we can combine the logarithms on the right side:
log(5x-4) = log((x+2) * 10^(1/3))
Since the logarithms on both sides are equal, the expressions inside the logarithms must also be equal:
5x - 4 = (x+2) * 10^(1/3)
Simplify the right side:
5x - 4 = (x+2) * (10^(1/3))
Now we need to solve for x. Start by distributing on the right side:
5x - 4 = (x+2) * (10^(1/3))
5x - 4 = x * (10^(1/3)) + 2 * (10^(1/3))
Now, solve for x by isolating the x term:
5x - x * (10^(1/3)) = 4 + 2 * (10^(1/3))
x(5 - 10^(1/3)) = 4 + 2 * (10^(1/3))
Divide both sides by (5 - 10^(1/3)):
x = (4 + 2 * (10^(1/3))) / (5 - 10^(1/3))
This is the simplified solution for x in the given equation.
To solve for x in the equation log(5x-4) = log(x+2) + 1/3, we can use the properties of logarithms.
First, we can start by simplifying the equation by combining the logarithmic terms:
log(5x-4) = log(x+2) + 1/3
Next, we can move the 1/3 term to the other side of the equation:
log(5x-4) - log(x+2) = 1/3
Using the logarithmic property log(a) - log(b) = log(a/b), we can rewrite the equation as:
log((5x-4)/(x+2)) = 1/3
Now, we can convert the logarithmic equation to an exponential equation. Remember that if log(b) = a, then b = 10^a.
So, we have:
(5x-4)/(x+2) = 10^(1/3)
To simplify further, we can cuberoot both sides:
[(5x-4)/(x+2)]^(1/3) = (10^(1/3))^(1/3)
Which simplifies to:
(5x-4)/(x+2) = 10^(1/9)
Now, we can cross-multiply to get rid of the fraction:
(5x-4) = (x+2) * 10^(1/9)
Expanding the right side of the equation:
5x - 4 = 10^(1/9) * x + 2 * 10^(1/9)
Now, let's isolate the terms with x:
5x - 10^(1/9) * x = 4 + 2 * 10^(1/9)
Combining like terms:
(5 - 10^(1/9)) * x = 4 + 2 * 10^(1/9)
Now, we can divide both sides by (5 - 10^(1/9)) to solve for x:
x = (4 + 2 * 10^(1/9)) / (5 - 10^(1/9))
This is the solution for x in the given equation.