Answer:
x = 6
Explanation:
Identifying log(x + 6), -log(x - 4), and log(x):
- log(x+6), - log(x - 4), and log(x) are common logs, which means their base is 10
- The 10 is usually invisible when dealing with common logs since it's assumed that we know we have a common log when we don't see another base).
Applying the quotient rule of logs:
- The quotient rule of logs states that the logarithm of a quotient is equal to a difference of logarithms.
- I'll provide a general example of this rule below using x and y as the arguments.
log(x / y) = log(x) - ln(y)
- As this example shows, we're able to got from the quotient to the difference (condensed to expanded) and from the difference back to the quotient (expanded to condensed).
Condensing log(x + 6) - log(x - 4):
Thus, we want to make (x + 6) and (x - 4) a quotient to get:
log ((x + 6) / (x - 4)) = log (x)
Solving for x:
- Now we can take the arguments outside the logs and work with them directly to solve for x:
((x + 6) / (x - 4)) = x
Multiplying both sides by x - 4 gives us:
(((x + 6) / (x - 4)) = x) * (x - 4)
x + 6 = x^2 - 4x
Putting the quadratic in standard form to solve for x:
- We want the quadratic to be in standard form to solve for x.
- The general equation of the standard form is given by:
ax^2 + bx + c = 0
Thus, we need to subtract x^2 and add 4x to both sides:
(x + 6 = x^2 - 4x) - x^2 + 4x
-x^2 + 5x + 6 = 0
Solving by factoring:
- For some quadratics, we're able to solve by factoring, which allows us to go from the standard to the factored form, whose general equation is given by:
- 0 = (x - p)(x - q)
- In the quadratic, -1 is our a value, 5 is our b value, and 6 is our c value.
- To solve by factoring, we want to find two numbers whose product equals a * c (-1 * 6 which equals -6) and whose sum equals b (5)
- The numbers -1 and 6 satisfy these criteria as -1 * 6 = -6 and -1 + 6 = 5.
- When using the factored form, we use the opposite sign of the two numbers we have.
Thus, the factored form is (x + 1)(x - 6) = 0
- Now we can solve for x by setting each term equal to 0:
Setting (x + 1) equal to 0:
(x + 1 = 0) - 1
x = -1
Setting (x - 6 = 0) + 6:
x = 6
- We can't have a negative answer when dealing with logs so the answer is x = 6.
Checking the validity of our answer:
- We can check our work by plugging in 6 for x and seeing if we get the same answer on both sides of the equation:
Plugging in 6 for x in log(x + 6) - log(x - 4) = log(x):
log(6 + 6) - log(6 - 4) = log(6)
log(12) - log(2) = log(6)
0.778 = 0.778
Thus, our answer is correct and x = 6.