Solve the following logarithmic equation. ln(x+3)-ln(x-2)=6
To solve the logarithmic equation ln(x+3) - ln(x-2) = 6, we can use logarithmic properties to simplify it.
First, we can combine the two logarithms using the quotient rule of logarithms. According to the quotient rule, ln(a) - ln(b) can be rewritten as ln(a/b). Applying this rule, we have:
ln((x+3)/(x-2)) = 6
Next, we can convert the equation to its exponential form. The exponential form of a logarithm states that if log base a of b equals c, then a raised to the power of c equals b. So, applying the exponential form to our equation, we have:
e^(ln((x+3)/(x-2))) = e^6
Where e is the base of the natural logarithm.
Simplifying further, we find:
(x+3)/(x-2) = e^6
Now, we can cross-multiply to eliminate the fraction:
(x+3) = e^6 * (x-2)
Expanding the right side of the equation, we have:
x + 3 = e^6 * x - 2e^6
Next, we can move all the terms containing x to one side of the equation and the constants to the other side:
x - e^6 * x = -2e^6 - 3
Factoring out x from the left side gives:
x(1 - e^6) = -2e^6 - 3
Finally, we can solve for x by dividing both sides of the equation by (1 - e^6):
x = (-2e^6 - 3) / (1 - e^6)
Hence, the solution to the logarithmic equation ln(x+3) - ln(x-2) = 6 is x = (-2e^6 - 3) / (1 - e^6).