If $$ \ln a=2, \ln b=3 $$, and $$ \ln c=5 $$, evaluate the following: (a) $$ \ln \left(\frac{a^{2}}{b^{2} c^{4}}\right)=\square $$ (b) $$ \ln \sqrt{a^{2} b^{2} c^{4}}=\square $$ Question 8

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**(a)**  
We start with the expression  
$$
\ln \left(\frac{a^{2}}{b^{2} c^{4}}\right)
$$
Using logarithm properties, we have:
$$
\ln \left(\frac{a^{2}}{b^{2} c^{4}}\right) = \ln (a^2) - \ln (b^2 c^4)
$$
Further,
$$
\ln (b^2 c^4) = \ln (b^2) + \ln (c^4)
$$
So, the expression becomes:
$$
\ln (a^2) - \ln (b^2) - \ln (c^4)
$$
Using the power rule of logarithms:
$$
2\ln a - 2\ln b - 4\ln c
$$
Substitute the given values $$\ln a = 2$$, $$\ln b = 3$$, and $$\ln c = 5$$:
$$
2(2) - 2(3) - 4(5) = 4 - 6 - 20 = -22
$$
Thus, the answer for (a) is $$-22$$.

---

**(b)**  
We begin with the expression:
$$
\ln \sqrt{a^{2} b^{2} c^{4}}
$$
Recall that taking the square root is equivalent to raising to the power $$\frac{1}{2}$$:
$$
\ln \sqrt{a^{2} b^{2} c^{4}} = \ln \left[(a^{2} b^{2} c^{4})^{\frac{1}{2}}\right]
$$
Using the power rule:
$$
= \frac{1}{2} \ln (a^{2} b^{2} c^{4})
$$
Now, expand the logarithm using properties:
$$
\ln (a^{2} b^{2} c^{4}) = \ln (a^2) + \ln (b^2) + \ln (c^4)
$$
Apply the power rule again:
$$
= 2\ln a + 2\ln b + 4\ln c
$$
Thus, the expression becomes:
$$
= \frac{1}{2}(2\ln a + 2\ln b + 4\ln c)
$$
Simplify:
$$
= \ln a + \ln b + 2\ln c
$$
Substitute the given values:
$$
= 2 + 3 + 2(5) = 2 + 3 + 10 = 15
$$
So, the answer for (b) is $$15$$.
**(a)** $$-22$$ **(b)** $$15$$

If , and , evaluate the following:
(a)
(b)
Question 8

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