how do i solve this equation?
log3x+log3(x-24)=4
To solve the equation log₃x + log₃(x - 24) = 4, you can use logarithmic properties to simplify it.
Step 1: Combine the logarithms using the product rule: logₐm + logₐn = logₐ(m x n). Applying this rule to the equation, we get:
log₃x(x - 24) = 4
Step 2: Convert the equation into exponential form. In logarithmic form, logₐb = c is equivalent to aᶜ = b. Applying this to the equation, we have:
3⁴ = x(x - 24)
Step 3: Simplify and solve for x. Expanding the exponential term, we get:
81 = x² - 24x
Step 4: Rearrange the equation into quadratic form. Combine the terms and set the equation equal to zero:
x² - 24x - 81 = 0
Step 5: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, factoring is not apparent, so we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Applying the formula, with a = 1, b = -24, and c = -81, we have:
x = (-(-24) ± √((-24)² - 4(1)(-81))) / (2(1))
Simplifying:
x = (24 ± √(576 + 324)) / 2
x = (24 ± √(900)) / 2
x = (24 ± 30) / 2
Now, we have two possible solutions:
x₁ = (24 + 30) / 2 = 54 / 2 = 27
x₂ = (24 - 30) / 2 = -6 / 2 = -3
Therefore, the solution to the equation log₃x + log₃(x - 24) = 4 is x = 27 or x = -3.
I will assume you mean log3 for all
log3x+log3(x-24)=4
log3 (x/(x-24)) = 4
3^4 = x/(x-24)
81 = x/(x-24)
81x - 1944 = x
x = 24.3