solve the equation
e^2 ln x - ln (x+2) = 8/3
wut
Hello I am here from the future
To solve the given equation, we need to first isolate the variable x. Here's how to do it step by step:
Step 1: Simplify the equation
e^2 ln x - ln (x+2) = 8/3
Step 2: Use logarithmic rules to rewrite the equation
ln(x^e^2) - ln(x+2) = 8/3
Step 3: Combine the logarithms into a single logarithm
ln(x^e^2 / (x+2)) = 8/3
Step 4: Take the exponential of both sides to eliminate the logarithm
e^(ln(x^e^2 / (x+2))) = e^(8/3)
Step 5: Simplify the left side of the equation using the exponent rule
x^e^2 / (x+2) = e^(8/3)
Step 6: Multiply both sides by (x+2) to eliminate the fraction
x^e^2 = (x + 2) * e^(8/3)
Step 7: Expand the right side of the equation
x^e^2 = x * e^(8/3) + 2 * e^(8/3)
Step 8: Rearrange the equation to have all terms on one side
x^e^2 - x * e^(8/3) - 2 * e^(8/3) = 0
At this point, we have a polynomial equation in terms of x with the unknown variable x. However, solving this equation for x algebraically is not trivial. We can use numerical methods such as Newton's method or a graphing calculator to find approximate solutions.
Alternatively, we can use a numerical solver or approximation method to find an approximate solution to this equation.
ln (x^e^2) - ln (x+2) = 2.667
ln [ (x^e^2)/(x+2) ] = 2.667
x^e^2 /(x+2) = 14.4
x^7.39/(x+2) = 14.4
I do not know how to proceed except numerically
x = 1, f = 1/3
x = 2, f = 41.9
x = 1.5, f = 5.71
x = 1.6, f = 8.95
x = 1.7, f = 13.64
x = 1.75, f = 16.57
x = 1.71, f = 14.2
that is close to 14.4, try x = 1.715 etc