To solve the equation 2log3(x-5) - 2 = log3(3x-5), we can use the properties of logarithms to simplify the equation.
First, let's break down the equation step by step:
2log3(x-5) - 2 = log3(3x-5) can be rewritten as:log3((x-5)^2) - log3(3) = log3(3x-5)
Combine the logarithms using the subtraction rule:log3((x-5)^2 / 3) = log3(3x-5)
Use the property that if loga(b) = loga(c), then b = c:(x-5)^2 / 3 = 3x - 5
Expand the left side of the equation and simplify:(x^2 - 10x + 25) / 3 = 3x - 5
Multiply both sides of the equation by 3 to get rid of the denominator:x^2 - 10x + 25 = 9x - 15
Rearrange the equation to set it equal to zero:x^2 - 19x + 40 = 0
Factor the quadratic equation:(x - 5)(x - 8) = 0
Solve for x by setting each factor to zero:x - 5 = 0, x - 8 = 0x = 5, x = 8
Therefore, the solutions to the equation 2log3(x-5) - 2 = log3(3x-5) are x = 5 and x = 8.
To solve the equation 2log3(x-5) - 2 = log3(3x-5), we can use the properties of logarithms to simplify the equation.
First, let's break down the equation step by step:
Use the power rule of logarithms:loga(b) - loga(c) = loga(b/c)
2log3(x-5) - 2 = log3(3x-5) can be rewritten as:
log3((x-5)^2) - log3(3) = log3(3x-5)
Combine the logarithms using the subtraction rule:
log3((x-5)^2 / 3) = log3(3x-5)
Use the property that if loga(b) = loga(c), then b = c:
(x-5)^2 / 3 = 3x - 5
Expand the left side of the equation and simplify:
(x^2 - 10x + 25) / 3 = 3x - 5
Multiply both sides of the equation by 3 to get rid of the denominator:
x^2 - 10x + 25 = 9x - 15
Rearrange the equation to set it equal to zero:
x^2 - 19x + 40 = 0
Factor the quadratic equation:
(x - 5)(x - 8) = 0
Solve for x by setting each factor to zero:
x - 5 = 0, x - 8 = 0
x = 5, x = 8
Therefore, the solutions to the equation 2log3(x-5) - 2 = log3(3x-5) are x = 5 and x = 8.