Question 7 If $$ \ln a=2, \ln b=3 $$, and $$ \ln c=5 $$, evaluate the following: (a) $$ \ln \left(\frac{a^{2}}{b^{2} c^{4}}\right)=\square $$ (b) $$ \ln \sqrt{a^{2} b^{2} c^{4}}= $$ Question 8 Solve the given equation for $$ x $$. $$ 7^{4 x-5}=32 $$ $$ x=\mathrm{P}^{2} $$ Question 9 Simplify the following into a single logarithm: $$ 5 \log (2)-2 \log (x) $$ O $$ \log \left(\frac{2^{5}}{x^{2}}\right) $$ O $$ \log (5 \cdot 2 \cdot 2 x) $$ O $$ \log \left(\frac{5 \cdot 2}{2 x}\right) $$ O $$ \log \left(5 \cdot 2 \cdot x^{2}\right) $$ O $$ \log \left(2^{5} x^{2}\right) $$

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**Question 7**

(a) We start by writing the logarithm using logarithm properties:
$$
\ln\left(\frac{a^2}{b^2 c^4}\right)=2\ln(a)-2\ln(b)-4\ln(c)
$$
Substitute $$\ln a=2$$, $$\ln b=3$$, and $$\ln c=5$$:
$$
2(2)-2(3)-4(5)=4-6-20=-22
$$
Thus, the answer is $$-22$$.

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(b) Express the square root as a power:
$$
\ln\sqrt{a^2b^2c^4}=\ln\left((a^2b^2c^4)^{\frac{1}{2}}\right)=\frac{1}{2}\ln(a^2b^2c^4)
$$
Using the logarithm properties:
$$
\frac{1}{2}\left(2\ln a+2\ln b+4\ln c\right)=\ln a+\ln b+2\ln c
$$
Substitute the given values:
$$
2+3+2(5)=2+3+10=15
$$
Thus, the answer is $$15$$.

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**Question 8**

We have the equation:
$$
7^{4x-5}=32
$$
Take the natural logarithm of both sides:
$$
\ln\left(7^{4x-5}\right)=\ln(32)
$$
Use the logarithm power rule:
$$
(4x-5)\ln(7)=\ln(32)
$$
Solve for $$ x $$:
$$
4x-5=\frac{\ln(32)}{\ln(7)}
$$
$$
4x=\frac{\ln(32)}{\ln(7)}+5
$$
$$
x=\frac{1}{4}\left(\frac{\ln(32)}{\ln(7)}+5\right)
$$
This expression is the solution for $$ x $$.

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**Question 9**

We simplify:
$$
5\log(2)-2\log(x)
$$
Rewrite each term using the power rule for logarithms:
$$
5\log(2)=\log(2^5) \quad \text{and} \quad -2\log(x)=\log(x^{-2})
$$
Combine using the product rule:
$$
\log(2^5x^{-2})=\log\left(\frac{2^5}{x^2}\right)
$$
Thus, the simplified single logarithm is:
$$
\log\left(\frac{2^5}{x^2}\right)
$$
**Question 7** (a) $$-22$$ (b) $$15$$ **Question 8** $$ x = \frac{1}{4}\left(\frac{\ln(32)}{\ln(7)} + 5\right) $$ **Question 9** $$ \log\left(\frac{2^5}{x^2}\right) $$

Question 7
If , and , evaluate the following:
(a)
(b)
Question 8
Solve the given equation for .

Question 9
Simplify the following into a single logarithm:

O
O
O
O
O

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Question 7 If , and , evaluate the following: (a) (b) Question 8 Solve the given equation for . Question 9 Simplify the following into a single logarithm: O O O O O