x lnx - 30 + 5 lnx - 6x = 0
Are you supposed to solve for x?
(5+x)ln x - 6x = 30
That looks to me like a transcendental equation that will have to be solved by iteration or graphing.
Try different values of x on the left side until you get +30.
If x = 0.01, (5+x)ln x - 6x = -23.12
If x = 0.05, (5+x)ln x - 6x = -15.4
If x = 0.1, (5+x)ln x - 6x = -12.3
If x = 1, (5+x)ln x - 6x = -6.0
If x = 2, (5+x)ln x - 6x = -7.15
If x = 10, (5+x)ln x - 6x = -25.6
If x = 20, (5+x)ln x - 6x = -45.1
It looks like there is no value of x for which the equation is valid.
To solve the given equation:
x lnx - 30 + 5 lnx - 6x = 0
Let's simplify it step by step:
Step 1: Combine like terms:
x lnx + 5 lnx - 6x - 30 = 0
Step 2: Group the terms involving lnx:
lnx (x + 5) - 6x - 30 = 0
Step 3: Move the terms without lnx to the other side:
lnx (x + 5) = 6x + 30
Step 4: Distribute lnx to the terms inside the parentheses:
x lnx + 5 lnx = 6x + 30
Step 5: Combine x lnx and 5 lnx using the rules of logarithm:
6 lnx = 6x + 30
Step 6: Divide both sides by 6:
lnx = x + 5
Now, we have a logarithmic equation. To solve for x, we can use the property of logarithms that states:
If ln(a) = ln(b), then a = b.
Applying this property, we can rewrite the equation as:
x + 5 = e^lnx
Step 7: Simplify the right side using the inverse of natural logarithm:
x + 5 = x
Step 8: Subtract x from both sides:
5 = 0
Since 5 is not equal to 0, we have a contradiction. Therefore, the given equation has no solution.