Write the expression as the logarithm of a single quantity.
ln 2 + 1/2 ln(x +1)-2 ln(1+ x^(1/2))
ln 2 + 1/2 ln(x +1)-2 ln(1+ x^(1/2))
= ln2 + ln √(x+1) - ln( 1+√x)^2
= ln [ 2√(x+1)/(1+√x)^2 ]
To write the expression as the logarithm of a single quantity, we can use the properties of logarithms. Let's simplify each term step by step.
1. ln 2:
Since there is no variable involved here, ln 2 can be written as ln(2).
2. 1/2 ln(x + 1):
The rule of logarithms that states log(A^n) = n * log(A) can be applied here. So, 1/2 ln(x + 1) can be rewritten as ln((x + 1)^(1/2)).
3. -2 ln(1 + x^(1/2)):
Using the same logarithmic rule, we can rewrite -2 ln(1 + x^(1/2)) as ln((1 + x^(1/2))^(-2)).
Putting it all together, the original expression can be written as:
ln(2) + ln((x + 1)^(1/2)) - ln((1 + x^(1/2))^2)
Now, we can simplify this further using log properties. The sum of logarithms can be written as the logarithm of their product, and the difference of logarithms can be written as the logarithm of their quotient.
Applying these rules, we have:
ln(2) + ln((x + 1)^(1/2)/((1 + x^(1/2))^2))
Finally, by combining the terms inside the logarithm into a single expression, we get:
ln(2(x + 1)^(1/2)/((1 + x^(1/2))^2))
Therefore, the expression can be written as ln(2(x + 1)^(1/2)/((1 + x^(1/2))^2)).