$$ \begin{array}{lllll} ext { 8. a) } \log _{2}\left(2^{m} ight) & ext { d) } \log _{4}\left(\frac{1}{4^{\frac{1}{n}}} ight) & ext { g) } \log _{b}\left(\frac{1}{b} ight) & ext { j) } \log _{a}\left(a^{n} ight) & ext { m) } \log _{a}\left(a^{\frac{1}{n}} ight) \ ext { b) } \log _{5}\left(\frac{1}{5^{m}} ight) & ext { e) } \log _{b}(b) & ext { h) } \log _{b}\left(b^{2} ight) & ext { k) } \log _{c}(\sqrt{c}) & ext { n) } \log _{a}\left(a^{\frac{m}{n}} ight) \ ext { c) } \log _{3}\left(3^{\frac{1}{n}} ight) & ext { f) } \log _{b}(1) & ext { i) } \log _{b}\left(\frac{1}{b^{2}} ight) & ext { l) } \log _{c}\left(\frac{1}{\sqrt{c}} ight) & ext { o) } \log _{a}\left(\frac{1}{a^{\frac{3}{5}}} ight)\end{array} $$

Question

$$ \begin{array}{lllll}	ext { 8. a) } \log _{2}\left(2^{m}ight) & 	ext { d) } \log _{4}\left(\frac{1}{4^{\frac{1}{n}}}ight) & 	ext { g) } \log _{b}\left(\frac{1}{b}ight) & 	ext { j) } \log _{a}\left(a^{n}ight) & 	ext { m) } \log _{a}\left(a^{\frac{1}{n}}ight) \ 	ext { b) } \log _{5}\left(\frac{1}{5^{m}}ight) & 	ext { e) } \log _{b}(b) & 	ext { h) } \log _{b}\left(b^{2}ight) & 	ext { k) } \log _{c}(\sqrt{c}) & 	ext { n) } \log _{a}\left(a^{\frac{m}{n}}ight) \ 	ext { c) } \log _{3}\left(3^{\frac{1}{n}}ight) & 	ext { f) } \log _{b}(1) & 	ext { i) } \log _{b}\left(\frac{1}{b^{2}}ight) & 	ext { l) } \log _{c}\left(\frac{1}{\sqrt{c}}ight) & 	ext { o) } \log _{a}\left(\frac{1}{a^{\frac{3}{5}}}ight)\end{array} $$

UpStudy AI · Accepted Answer

To solve the logarithmic expressions provided, we will use the properties of logarithms. Here are the relevant properties:

1. $$ \log_b(b^x) = x $$
2. $$ \log_b(1) = 0 $$
3. $$ \log_b\left(\frac{1}{b^x}\right) = -x $$
4. $$ \log_b\left(b^{\frac{1}{n}}\right) = \frac{1}{n} $$
5. $$ \log_b\left(\sqrt{b}\right) = \frac{1}{2} $$

Now, let's evaluate each expression step by step:

### a) $$ \log_{2}(2^{m}) $$
Using property 1:
$$
\log_{2}(2^{m}) = m
$$

### b) $$ \log_{5}\left(\frac{1}{5^{m}}\right) $$
Using property 3:
$$
\log_{5}\left(\frac{1}{5^{m}}\right) = -m
$$

### c) $$ \log_{3}\left(3^{\frac{1}{n}}\right) $$
Using property 1:
$$
\log_{3}\left(3^{\frac{1}{n}}\right) = \frac{1}{n}
$$

### d) $$ \log_{4}\left(\frac{1}{4^{\frac{1}{n}}}\right) $$
Using property 3:
$$
\log_{4}\left(\frac{1}{4^{\frac{1}{n}}}\right) = -\frac{1}{n}
$$

### e) $$ \log_{b}(b) $$
Using property 1:
$$
\log_{b}(b) = 1
$$

### f) $$ \log_{b}(1) $$
Using property 2:
$$
\log_{b}(1) = 0
$$

### g) $$ \log_{b}\left(\frac{1}{b}\right) $$
Using property 3:
$$
\log_{b}\left(\frac{1}{b}\right) = -1
$$

### h) $$ \log_{b}\left(b^{2}\right) $$
Using property 1:
$$
\log_{b}\left(b^{2}\right) = 2
$$

### i) $$ \log_{b}\left(\frac{1}{b^{2}}\right) $$
Using property 3:
$$
\log_{b}\left(\frac{1}{b^{2}}\right) = -2
$$

### j) $$ \log_{a}\left(a^{n}\right) $$
Using property 1:
$$
\log_{a}\left(a^{n}\right) = n
$$

### k) $$ \log_{c}(\sqrt{c}) $$
Using property 4:
$$
\log_{c}(\sqrt{c}) = \frac{1}{2}
$$

### l) $$ \log_{c}\left(\frac{1}{\sqrt{c}}\right) $$
Using property 3:
$$
\log_{c}\left(\frac{1}{\sqrt{c}}\right) = -\frac{1}{2}
$$

### m) $$ \log_{a}\left(a^{\frac{1}{n}}\right) $$
Using property 1:
$$
\log_{a}\left(a^{\frac{1}{n}}\right) = \frac{1}{n}
$$

### n) $$ \log_{a}\left(a^{\frac{m}{n}}\right) $$
Using property 1:
$$
\log_{a}\left(a^{\frac{m}{n}}\right) = \frac{m}{n}
$$

### o) $$ \log_{a}\left(\frac{1}{a^{\frac{3}{5}}}\right) $$
Using property 3:
$$
\log_{a}\left(\frac{1}{a^{\frac{3}{5}}}\right) = -\frac{3}{5}
$$

### Summary of Results:
- a) $$ m $$
- b) $$ -m $$
- c) $$ \frac{1}{n} $$
- d) $$ -\frac{1}{n} $$
- e) $$ 1 $$
- f) $$ 0 $$
- g) $$ -1 $$
- h) $$ 2 $$
- i) $$ -2 $$
- j) $$ n $$
- k) $$ \frac{1}{2} $$
- l) $$ -\frac{1}{2} $$
- m) $$ \frac{1}{n} $$
- n) $$ \frac{m}{n} $$
- o) $$ -\frac{3}{5} $$
Here are the simplified results for each logarithmic expression: - a) $$ m $$ - b) $$ -m $$ - c) $$ \frac{1}{n} $$ - d) $$ -\frac{1}{n} $$ - e) $$ 1 $$ - f) $$ 0 $$ - g) $$ -1 $$ - h) $$ 2 $$ - i) $$ -2 $$ - j) $$ n $$ - k) $$ \frac{1}{2} $$ - l) $$ -\frac{1}{2} $$ - m) $$ \frac{1}{n} $$ - n) $$ \frac{m}{n} $$ - o) $$ -\frac{3}{5} $$

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