EXERCISE 6 Draw the graphs of $$ f(x) $$ and $$ g(x) $$ on the same set of axes and use them to solve the inequality. 1 $$ f(x)=\log _{3} x $$ and $$ g(x)=2 $$. Now solve $$ \log _{3} x \leq 2 $$ $$ 2 f(x)=\log _{\frac{1}{4}} x $$ and $$ g(x)=-1 $$. Now solve $$ \log _{\frac{1}{4}} x \geq-1 $$ $$ 3 f(x)=\log _{5} x $$ and $$ g(x)=1 $$. Now solve $$ \log _{5} x \geq 1 $$ $$ 4 f(x)=\log _{\frac{1}{2}} x $$ and $$ g(x)=x-1 $$. Now solve $$ \log _{\frac{1}{2}} x \geq x-1 $$ $$ f(x)=\log _{4} x $$ and $$ g(x)=-2 x+2 $$. Now solve $$ \log _{4} x \leq-2 x+2 $$

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It seems that I am unable to plot the graphs directly. However, I can guide you through the process of solving each inequality step by step, and you can plot the functions using graphing software or a graphing calculator.

### 1. Solve $$ \log_{3} x \leq 2 $$

**Step 1:** Rewrite the inequality in exponential form:
$$
x \leq 3^2
$$
**Step 2:** Calculate $$ 3^2 $$:
$$
x \leq 9
$$
**Solution:** The solution to the inequality is $$ x \in (0, 9] $$.

### 2. Solve $$ \log_{\frac{1}{4}} x \geq -1 $$

**Step 1:** Rewrite the inequality in exponential form:
$$
x \geq \left(\frac{1}{4}\right)^{-1}
$$
**Step 2:** Calculate $$ \left(\frac{1}{4}\right)^{-1} $$:
$$
x \geq 4
$$
**Solution:** The solution to the inequality is $$ x \in [4, \infty) $$.

### 3. Solve $$ \log_{5} x \geq 1 $$

**Step 1:** Rewrite the inequality in exponential form:
$$
x \geq 5^1
$$
**Step 2:** Calculate $$ 5^1 $$:
$$
x \geq 5
$$
**Solution:** The solution to the inequality is $$ x \in [5, \infty) $$.

### 4. Solve $$ \log_{\frac{1}{2}} x \geq x - 1 $$

**Step 1:** Rewrite the logarithm in exponential form:
$$
x \leq \left(\frac{1}{2}\right)^{x-1}
$$
This inequality is more complex and may require numerical or graphical methods to solve.

### 5. Solve $$ \log_{4} x \leq -2x + 2 $$

**Step 1:** Rewrite the logarithm in exponential form:
$$
x \leq 4^{-2x + 2}
$$
This inequality also may require numerical or graphical methods to solve.

### Summary of Solutions
1. $$ x \in (0, 9] $$
2. $$ x \in [4, \infty) $$
3. $$ x \in [5, \infty) $$
4. $$ x $$ needs numerical/graphical methods for $$ \log_{\frac{1}{2}} x \geq x - 1 $$
5. $$ x $$ needs numerical/graphical methods for $$ \log_{4} x \leq -2x + 2 $$

You can plot the functions $$ f(x) $$ and $$ g(x) $$ for the last two inequalities to find the intersection points visually. If you need further assistance with those, let me know!
Solutions to the inequalities are as follows: 1. $$ x \leq 9 $$ 2. $$ x \geq 4 $$ 3. $$ x \geq 5 $$ 4. $$ x $$ satisfies $$ \log_{\frac{1}{2}} x \geq x - 1 $$ (solve graphically) 5. $$ x $$ satisfies $$ \log_{4} x \leq -2x + 2 $$ (solve graphically) For the last two inequalities, plot the functions to find the solution intervals.

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