1/2ln(x+3)-lnx=0
To solve the equation 1/2 ln(x+3) - ln(x) = 0, we can follow these steps:
Step 1: Combine the logarithms using the quotient rule of logarithms. According to the quotient rule, ln(a) - ln(b) = ln(a/b). Applying this rule to our equation, we get:
ln((x+3)/x) = 0
Step 2: Raise both sides of the equation as a power of e (since ln and e^x are inverse functions). By doing this, we can eliminate the natural logarithm:
e^(ln((x+3)/x)) = e^0
(x+3)/x = 1
Step 3: Cross-multiply to get rid of the fraction:
x + 3 = x
Step 4: Subtract x from both sides of the equation:
3 = 0
Step 5: Since the equation 3 = 0 is impossible, it means that there is no solution to the original equation 1/2 ln(x+3) - ln(x) = 0.
Therefore, the equation has no solution.