Solve
log(5x-1) = 2 + log(x-2)
log(5x-1) = 2 + log(x-2)
log(5x-1) - log(x-2) = 2
log ( (5x-1)/(x-2) ) = 2
(5x-1)/(x-2) = 10^2 = 100
100x - 200 = 5x-1
95x = 199
x = 199/95
How did you arrive at 10^2?
By the definition of logs
loga b = c <------> a^c = b
so if log10( (5x-1)/(x-2) ) = 2 , then
10^2 = ( (5x-1)/(x-2) )
To solve the equation log(5x-1) = 2 + log(x-2), we'll use the properties of logarithms. Here's the step-by-step procedure:
Step 1: Combine the logarithms using the addition property of logarithms. According to this property, log(a) + log(b) = log(a * b).
log(5x - 1) = log((x - 2) * 10^2)
Step 2: Since both sides of the equation have the same logarithm, we can cancel them out, resulting in:
5x - 1 = (x - 2) * 10^2
Step 3: Simplify both sides of the equation:
5x - 1 = (x - 2) * 100
Step 4: Expand the right side of the equation:
5x - 1 = 100x - 200
Step 5: Move all the x terms to one side of the equation and the constants to the other side:
5x - 100x = -200 + 1
-95x = -199
Step 6: Divide both sides of the equation by -95 to solve for x:
x = -199 / -95
Step 7: Simplify the result:
x ≈ 2.0947
Therefore, the solution to the equation log(5x-1) = 2 + log(x-2) is x ≈ 2.0947.