Question 7 If $$ \ln a=2, \ln b=3 $$, and $$ \ln c=5 $$, evaluate the following: (a) $$ \ln \left(\frac{a^{2}}{b^{2} c^{4}}\right)=\square 1 \mathrm{pt} $$ (b) $$ \ln \sqrt{a^{2} b^{2} c^{4}}=\square $$ Question 8

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**Step-by-step solution:**

**(a)**  
We need to evaluate  
$$
\ln\left(\frac{a^2}{b^2 c^4}\right).
$$

Using the properties of logarithms, we have  
$$
\ln\left(\frac{a^2}{b^2 c^4}\right) = \ln(a^2) - \ln(b^2 c^4).
$$  
Further apply the logarithmic properties:  
$$
\ln(a^2) = 2\ln a,
$$  
$$
\ln(b^2 c^4) = \ln(b^2) + \ln(c^4) = 2\ln b + 4\ln c.
$$

Substitute these into the expression:  
$$
\ln\left(\frac{a^2}{b^2 c^4}\right) = 2\ln a - 2\ln b - 4\ln c.
$$

Given that  
$$
\ln a = 2,\quad \ln b = 3,\quad \ln c = 5,
$$  
we substitute these values:  
$$
2(2) - 2(3) - 4(5) = 4 - 6 - 20.
$$

Simplify the numerical expression:  
$$
4 - 6 - 20 = -22.
$$

Thus, the answer for part (a) is  
$$
-22.
$$

**(b)**  
We need to evaluate  
$$
\ln \sqrt{a^2 b^2 c^4}.
$$

First, rewrite the square root as an exponent:  
$$
\sqrt{a^2 b^2 c^4} = (a^2 b^2 c^4)^{\frac{1}{2}}.
$$

Take the logarithm:  
$$
\ln\left((a^2 b^2 c^4)^{\frac{1}{2}}\right) = \frac{1}{2}\ln(a^2 b^2 c^4).
$$

Apply the logarithm properties to the product:  
$$
\ln(a^2 b^2 c^4) = \ln(a^2) + \ln(b^2) + \ln(c^4) = 2\ln a + 2\ln b + 4\ln c.
$$

Thus,  
$$
\ln \sqrt{a^2 b^2 c^4} = \frac{1}{2}(2\ln a + 2\ln b + 4\ln c).
$$
Simplify by factoring the $$\frac{1}{2}$$:  
$$
= \ln a + \ln b + 2\ln c.
$$

Now substitute the given values:  
$$
\ln a = 2,\quad \ln b = 3,\quad \ln c = 5.
$$
So,  
$$
\ln a + \ln b + 2\ln c = 2 + 3 + 2(5) = 2 + 3 + 10.
$$
Simplify further:  
$$
2 + 3 + 10 = 15.
$$

Thus, the answer for part (b) is  
$$
15.
$$
**(a) -22 (b) 15**

Question 7
If , and , evaluate the following:
(a)
(b)
Question 8

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Question 7 If , and , evaluate the following: (a) (b) Question 8