Final answer:
The expression 'ln 9 + 1/2 ln(x + 1) − 4 ln(1 + √x)' can be simplified to a single logarithm by applying the power rule and then combining terms using the product and quotient rules of logarithms, resulting in 'ln(9 × (x + 1)^{1/2} / (1 + √x)^{4})'.
Step-by-step explanation:
To write the expression as a logarithm of a single quantity, we can use the properties of logarithms. Specifically, we'll use the product rule, the quotient rule, and the power rule for logarithms. The given expression is:
ln 9 + 1/2 ln(x + 1) − 4 ln(1 + √x)
First, apply the power rule, which allows us to move the coefficients of the logarithms up as exponents of their arguments:
ln 9 + ln(x + 1)1/2 − ln(1 + √x)4
Next, using the product and quotient rules, we combine these into a single logarithm:
ln(9 × (x + 1)1/2 / (1 + √x)4)
This is the desired expression as a logarithm of a single quantity.