$$ \begin{array}{lll} ext { Simplify: } & ext { b } \log _{t}\left(\frac{1}{t} ight) & ext { c } \log _{x} \sqrt{x} \ \begin{array}{ll} ext { a } \log _{x}\left(x^{2} ight) & ext { e } \log _{k} \sqrt[4]{k} \ ext { d } \log _{m}\left(m^{3} ight) & ext { h } \log _{a}\left(\frac{1}{\sqrt{a}} ight)\end{array} & ext { f } \log _{x}(x \sqrt{x}) \ ext { i } \log _{a}\left(\frac{1}{a^{2}} ight) & ext { i } \log _{m} \sqrt{m^{5}}\end{array} $$

Question

$$ \begin{array}{lll}	ext { Simplify: } & 	ext { b } \log _{t}\left(\frac{1}{t}ight) & 	ext { c } \log _{x} \sqrt{x} \ \begin{array}{ll}	ext { a } \log _{x}\left(x^{2}ight) & 	ext { e } \log _{k} \sqrt[4]{k} \ 	ext { d } \log _{m}\left(m^{3}ight) & 	ext { h } \log _{a}\left(\frac{1}{\sqrt{a}}ight)\end{array} & 	ext { f } \log _{x}(x \sqrt{x}) \ 	ext { i } \log _{a}\left(\frac{1}{a^{2}}ight) & 	ext { i } \log _{m} \sqrt{m^{5}}\end{array} $$

UpStudy AI · Accepted Answer

Let's simplify each logarithmic expression step by step.

1. **For $$ b \log_{t}\left(\frac{1}{t}\right) $$**:
   $$
   \log_{t}\left(\frac{1}{t}\right) = \log_{t}(t^{-1}) = -1
   $$
   Thus, $$ b \log_{t}\left(\frac{1}{t}\right) = -b $$.

2. **For $$ c \log_{x} \sqrt{x} $$**:
   $$
   \log_{x} \sqrt{x} = \log_{x} (x^{1/2}) = \frac{1}{2}
   $$
   Thus, $$ c \log_{x} \sqrt{x} = \frac{c}{2} $$.

3. **For $$ a \log_{x}\left(x^{2}\right) $$**:
   $$
   \log_{x}\left(x^{2}\right) = 2
   $$
   Thus, $$ a \log_{x}\left(x^{2}\right) = 2a $$.

4. **For $$ e \log_{k} \sqrt[4]{k} $$**:
   $$
   \log_{k} \sqrt[4]{k} = \log_{k} (k^{1/4}) = \frac{1}{4}
   $$
   Thus, $$ e \log_{k} \sqrt[4]{k} = \frac{e}{4} $$.

5. **For $$ d \log_{m}\left(m^{3}\right) $$**:
   $$
   \log_{m}\left(m^{3}\right) = 3
   $$
   Thus, $$ d \log_{m}\left(m^{3}\right) = 3d $$.

6. **For $$ h \log_{a}\left(\frac{1}{\sqrt{a}}\right) $$**:
   $$
   \log_{a}\left(\frac{1}{\sqrt{a}}\right) = \log_{a}(a^{-1/2}) = -\frac{1}{2}
   $$
   Thus, $$ h \log_{a}\left(\frac{1}{\sqrt{a}}\right) = -\frac{h}{2} $$.

7. **For $$ f \log_{x}(x \sqrt{x}) $$**:
   $$
   \log_{x}(x \sqrt{x}) = \log_{x}(x^{3/2}) = \frac{3}{2}
   $$
   Thus, $$ f \log_{x}(x \sqrt{x}) = \frac{3f}{2} $$.

8. **For $$ i \log_{a}\left(\frac{1}{a^{2}}\right) $$**:
   $$
   \log_{a}\left(\frac{1}{a^{2}}\right) = \log_{a}(a^{-2}) = -2
   $$
   Thus, $$ i \log_{a}\left(\frac{1}{a^{2}}\right) = -2i $$.

9. **For $$ i \log_{m} \sqrt{m^{5}} $$**:
   $$
   \log_{m} \sqrt{m^{5}} = \log_{m} (m^{5/2}) = \frac{5}{2}
   $$
   Thus, $$ i \log_{m} \sqrt{m^{5}} = \frac{5i}{2} $$.

Now, let's summarize the simplified results:

- $$ b \log_{t}\left(\frac{1}{t}\right) = -b $$
- $$ c \log_{x} \sqrt{x} = \frac{c}{2} $$
- $$ a \log_{x}\left(x^{2}\right) = 2a $$
- $$ e \log_{k} \sqrt[4]{k} = \frac{e}{4} $$
- $$ d \log_{m}\left(m^{3}\right) = 3d $$
- $$ h \log_{a}\left(\frac{1}{\sqrt{a}}\right) = -\frac{h}{2} $$
- $$ f \log_{x}(x \sqrt{x}) = \frac{3f}{2} $$
- $$ i \log_{a}\left(\frac{1}{a^{2}}\right) = -2i $$
- $$ i \log_{m} \sqrt{m^{5}} = \frac{5i}{2} $$

These are the simplified forms of the given logarithmic expressions.
Here are the simplified forms of the logarithmic expressions: - $$ b \log_{t}\left(\frac{1}{t}\right) = -b $$ - $$ c \log_{x} \sqrt{x} = \frac{c}{2} $$ - $$ a \log_{x}\left(x^{2}\right) = 2a $$ - $$ e \log_{k} \sqrt[4]{k} = \frac{e}{4} $$ - $$ d \log_{m}\left(m^{3}\right) = 3d $$ - $$ h \log_{a}\left(\frac{1}{\sqrt{a}}\right) = -\frac{h}{2} $$ - $$ f \log_{x}(x \sqrt{x}) = \frac{3f}{2} $$ - $$ i \log_{a}\left(\frac{1}{a^{2}}\right) = -2i $$ - $$ i \log_{m} \sqrt{m^{5}} = \frac{5i}{2} $$

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