Solve the lograthmic function:
ln(X+5) = ln(X-1) - ln(X+1)
The answer has no solution but I do not know how to come to that answer. Help is greatly appreciated!
ln(x+5) = ln[(x-1)/(x+1)]
x+5 = (x-1)/(x+1)
(x+5)(x+1) = x-1
x^2 + 6x + 5 = x-1
x^2 + 5x + 6 = 0
(x+2)(x+3) = 0
x = -2 or x=-3
but in ln(x-1), x-1>0 or x > 1
remember we can't take the log of a negative number and we would get ln(-3) and ln(-4) which would be undefined.
so neither of the solutions are valid, and
there is no solution to your equation.
Thank you soo much!
To solve the logarithmic equation ln(X+5) = ln(X-1) - ln(X+1), we can start by simplifying the equation using logarithmic properties.
1. Using the property ln(a) - ln(b) = ln(a/b), we can rewrite the equation as ln(X+5) = ln((X-1)/(X+1)).
2. Now, we can eliminate the natural logarithm on both sides of the equation by taking the antilogarithm (the exponential function e^x) of both sides. This will result in e^ln(X+5) = e^ln((X-1)/(X+1)).
3. The antilogarithm of ln(x) is simply x. So, the equation becomes e^(ln(X+5)) = e^(ln((X-1)/(X+1))).
4. Simplifying further, we have X+5 = (X-1)/(X+1).
5. To eliminate the fraction, we can cross-multiply, leading to (X+5)(X+1) = X-1.
6. Expanding the left side, we get X^2 + 6X + 5 = X - 1.
7. Combining like terms, we have X^2 + 6X - X + 5 + 1 = 0.
8. Simplifying further, we get X^2 + 5X + 6 = 0.
At this point, we have a quadratic equation. However, if we try to solve the equation using factoring, quadratic formula, or completing the square, we can see that there are no real or complex solutions. Therefore, the equation ln(X+5) = ln(X-1) - ln(X+1) has no solution.