Solve the equation and check your solutions. Note any extraneous answers.
log(3x-13)=2 - log x
assuming base 10,
log(3x-13) = log100 - logx
log(3x-13) = log(100/x)
3x-13 = 100/x
3x^2 - 13x - 100 = 0
(3x-25)(x+4) = 0
x = 25/3, since x = -4 is not allowed.
To solve this equation, we'll need to use properties of logarithms.
Step 1: Combine the two logarithms on the right side using the quotient property.
log(3x-13) = log(10^2/x) (since log x = log(10^x))
Step 2: Apply the power rule to get rid of the logarithms on both sides.
3x - 13 = 10^2/x (since log a = b is equivalent to saying 10^b = a)
Step 3: Multiply both sides of the equation by x to eliminate the fraction on the right side.
x(3x - 13) = 100
Step 4: Expand and simplify the equation.
3x^2 - 13x = 100
Step 5: Rearrange the equation in standard quadratic form.
3x^2 - 13x - 100 = 0
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 3, b = -13, and c = -100.
Substituting these values into the quadratic formula, we have:
x = (-(-13) ± √((-13)^2 - 4(3)(-100))) / (2(3))
x = (13 ± √(169 + 1200)) / 6
x = (13 ± √1369) / 6
Now, let's simplify and evaluate both solutions:
For x = (13 + √1369) / 6:
x = (13 + 37) / 6
x = 50 / 6
x = 25/3
For x = (13 - √1369) / 6:
x = (13 - 37) / 6
x = -24 / 6
x = -4
Now, we need to check if these solutions are valid by substituting them back into the original equation and verifying that both sides are equal.
Checking x = 25/3:
log(3(25/3)-13) = 2 - log(25/3)
log(25 - 13) = 2 - log(25/3)
log(12) = 2 - log(25/3)
log(12) = 2 - (log(25) - log(3))
log(12) = 2 - (log(5^2) - log(3))
log(12) = 2 - (2log(5) - log(3))
Using log properties:
log(12) = 2 - 2log(5) + log(3)
log(12) = -2log(5) + log(3) + 2
Now, evaluate each side:
log(12) ≈ 1.07918
-2log(5) + log(3) + 2 ≈ 1.07918
Since both sides are approximately equal, the solution x = 25/3 is valid.
Checking x = -4:
log(3(-4)-13) = 2 - log(-4)
log(-12) ≠ 2 - log(-4)
Here, we encounter a problem. The logarithm of a negative number is undefined, so substituting x = -4 into the original equation leads to an extraneous solution.
Therefore, the only valid solution to the equation is x = 25/3.