Write as a single logarithm of a single quantity
ln(3)+1/2ln(x+2)-4ln(1+sqrtx)
(1/2) ln (x+2) maybe ?
that would be ln sqrt (x+2)
so
ln [ 3 sqrt(x+2)/(1+sqrt x)^4 ]
Is the entire problem to the fourth power, or just the denominator?
Denominator
To express the given expression as a single logarithm of a single quantity, we can utilize several logarithmic properties and simplify the expression step by step.
The given expression is:
ln(3) + 1/2 ln(x+2) - 4 ln(1+sqrt(x))
Let's work on simplifying each term individually:
1. ln(3): ln(3) is already a single logarithm, so we can leave it as it is.
2. 1/2 ln(x+2): We can use the power property of logarithms to rewrite 1/2 ln(x+2) as ln((x+2)^(1/2)).
3. -4 ln(1+sqrt(x)): Similarly, we can rewrite -4 ln(1+sqrt(x)) as ln((1+sqrt(x))^(-4)), using the power property of logarithms.
By applying the above simplifications, the expression becomes:
ln(3) + ln((x+2)^(1/2)) - ln((1+sqrt(x))^(-4))
Now, we can simplify further by combining the logarithms using the properties of addition and subtraction of logarithms:
ln(3) + ln((x+2)^(1/2)) - ln((1+sqrt(x))^(-4))
Using the property of subtraction of logarithms, we take the difference of the second and third term:
ln(3) + ln((x+2)^(1/2)/(1+sqrt(x))^(-4))
To simplify it even further, we can use the property of division of logarithms:
ln(3 * (x+2)^(1/2)/(1+sqrt(x))^(-4))
Finally, we have expressed the given expression as a single logarithm of a single quantity:
ln(3 * (x+2)^(1/2)/(1+sqrt(x))^(-4))