The answer to the inequality log10(x2-7x)<log10(3-x)+log102 is
log(x^2-7x)<log(3-x)+log2 ,
from definition of logs, initial restrictions: x > 7 , x ≠0 , x < 3
let's just look at each part
clearly log(3-x) + log2
= log(6-2x) is only defined for x < 3
and log(x^2 - 7x) means that x^2 - 7x > 0
x(x - 7) > 0 , so x < 0 or x > 7
but in conjunction with the first restriction , we have x < 0 so far.
ok, look at
log(x^2 - 7x) = log(2(3-x))
log(x^2 - 7x) = log (6 - 2x)
antilog both sides:
x^2 - 7x = 6 - 2x
x^2 - 5x - 6 = 0
(x - 6)(x + 1) = 0
x = -1 or x = 6, but we know x < 0
they intersect at x = -1
check for a value less than -1,
let x = -5
log(25 + 35) < log(6 + 10)
log(60) < log 16 , which is false , so x is NOT < -1
check with a value between -1 and 0
try x = -.5
log(.25 + 3.5) < log(6 - 1)
log 3.75 < log 5, which is TRUE.
so ......
-1 < x < 0
I graphed y = log(x^2 - 7x) and y = log(6-2x) on Desmos
and it shows my answer is correct.
The answer to the inequality log(x^2-7x)<log(3-x)+log2 is
To solve the inequality log10(x^2 - 7x) < log10(3 - x) + log10(2), we can follow these steps:
Step 1: Combine the logarithms on the right side of the inequality.
- Using the rule log(a) + log(b) = log(ab), we can rewrite the inequality as: log10(x^2 - 7x) < log10((3 - x) * 2).
Step 2: Convert the inequality to exponential form.
- The logarithmic inequality log10(a) < log10(b) can be rewritten in exponential form as a < b.
- Applying this rule, we have: x^2 - 7x < (3 - x) * 2.
Step 3: Simplify the inequality and solve for x.
- Distributing the right side of the inequality, we get: x^2 - 7x < 6 - 2x.
- Rearranging terms, we have: x^2 - 5x - 6 < 0.
- Factoring the quadratic expression, we have: (x - 6)(x + 1) < 0.
- Now, we have two cases to consider:
Case 1: (x - 6) < 0 and (x + 1) > 0.
- Solving these inequalities separately, we get: x < 6 and x > -1.
- However, these inequalities cannot be satisfied simultaneously since there is no overlap in the solution range.
Case 2: (x - 6) > 0 and (x + 1) < 0.
- Solving these inequalities separately, we get: x > 6 and x < -1.
- The solution to this case is x < -1 and x > 6.
- However, this case also does not have a valid solution since there is no overlap in the solution range.
Conclusion: After analyzing both cases, we find that there is no solution that satisfies the inequality log10(x^2 - 7x) < log10(3 - x) + log10(2). Therefore, the inequality has no solution.