A(1,0) B $$ (0.1) $$ L $$ \left(\frac{1}{2},-1 ight) $$ D (2.1) 21. What is the solution set of the equation $$ \left(\frac{4}{23} ight)^{3^{2}}=\left(\frac{5}{2} ight)^{3 n+a} $$ ? A [1] Is $$ |-1| $$ ( $$ \left[\frac{1}{2} ight\} $$ D. 0 22. A function is said to be an exponental function, where A. $$ f(x)=a^{2}, a c \lambda $$ and $$ a=1 $$ $$ f(x)=a t, a0 $$ and $$ a
eq 1 $$ B $$ f(x)=a x, a \geq 0 $$ D $$ f(x)=a^{\prime}, a0 $$ 23. The solution of the equation $$ (3 \cdot)^{2} 5 x-3 \sqrt{5} $$ is: A $$ \frac{1}{2} $$ B $$ -\frac{1}{4} $$ C. $$ \frac{1}{45} $$ D $$ \frac{1}{2} $$ 24. What is the value of xthat makes $$ 2^{2 x},-37 $$ A $$ x=\frac{1}{2}\left(\log _{2} 3 ight)+1 $$ $$ B x=\frac{1}{2}+\log _{2} 3 $$ [ $$ x=\log 3+\log 2 $$ $$ D x=\frac{1}{2}\left(1+\log _{2} 3 ight) $$

Question

A(1,0) B $$ (0.1) $$ L $$ \left(\frac{1}{2},-1ight) $$ D (2.1) 21. What is the solution set of the equation $$ \left(\frac{4}{23}ight)^{3^{2}}=\left(\frac{5}{2}ight)^{3 n+a} $$ ? A [1] Is $$ |-1| $$ ( $$ \left[\frac{1}{2}ight\} $$ D. 0 22. A function is said to be an exponental function, where A. $$ f(x)=a^{2}, a c \lambda $$ and $$ a=1 $$ $$ f(x)=a t, a>0 $$ and $$ a 
eq 1 $$ B $$ f(x)=a x, a \geq 0 $$ D $$ f(x)=a^{\prime}, a>0 $$ 23. The solution of the equation $$ (3 \cdot)^{2} 5 x-3 \sqrt{5} $$ is: A $$ \frac{1}{2} $$ B $$ -\frac{1}{4} $$ C. $$ \frac{1}{45} $$ D $$ \frac{1}{2} $$ 24. What is the value of xthat makes $$ 2^{2 x},-37 $$ A $$ x=\frac{1}{2}\left(\log _{2} 3ight)+1 $$ $$ B x=\frac{1}{2}+\log _{2} 3 $$ [ $$ x=\log 3+\log 2 $$ $$ D x=\frac{1}{2}\left(1+\log _{2} 3ight) $$

UpStudy AI · Accepted Answer

Let's break down the problems step by step.

### Problem 21
We need to solve the equation:

$$
\left(\frac{4}{23}\right)^{3^{2}}=\left(\frac{5}{2}\right)^{3 n+a}
$$

1. **Simplify the left side**:
   $$
   3^2 = 9 \implies \left(\frac{4}{23}\right)^{9}
   $$

2. **Set the equation**:
   $$
   \left(\frac{4}{23}\right)^{9} = \left(\frac{5}{2}\right)^{3n + a}
   $$

3. **Taking logarithm on both sides**:
   $$
   9 \log\left(\frac{4}{23}\right) = (3n + a) \log\left(\frac{5}{2}\right)
   $$

4. **Solving for $$ n $$**:
   $$
   3n + a = \frac{9 \log\left(\frac{4}{23}\right)}{\log\left(\frac{5}{2}\right)}
   $$
   $$
   n = \frac{1}{3}\left(\frac{9 \log\left(\frac{4}{23}\right)}{\log\left(\frac{5}{2}\right)} - a\right)
   $$

The solution set depends on the values of $$ a $$ and $$ n $$. Without specific values for $$ a $$, we cannot determine a numerical solution.

### Problem 22
A function is said to be an exponential function if it can be expressed in the form:

$$
f(x) = a^x \quad \text{where } a > 0 \text{ and } a \neq 1
$$

Thus, the correct answer is:
- **B**: $$ f(x) = a^t, a > 0 \text{ and } a \neq 1 $$

### Problem 23
We need to solve the equation:

$$
(3 \cdot)^{2} 5 x - 3 \sqrt{5} = 0
$$

This equation seems to be incomplete or incorrectly formatted. Assuming it should be:

$$
(3^2) \cdot 5x - 3\sqrt{5} = 0
$$

1. **Calculate $$ 3^2 $$**:
   $$
   9 \cdot 5x - 3\sqrt{5} = 0
   $$
   $$
   45x = 3\sqrt{5}
   $$
   $$
   x = \frac{3\sqrt{5}}{45} = \frac{\sqrt{5}}{15}
   $$

None of the provided options match this solution, indicating a possible error in the problem statement.

### Problem 24
We need to solve for $$ x $$ in the equation:

$$
2^{2x} = -37
$$

Since $$ 2^{2x} $$ is always positive, there is no real solution for this equation. Thus, the value of $$ x $$ that makes this equation true does not exist in the real number system.

### Summary of Answers
- **21**: Cannot determine without values for $$ a $$ and $$ n $$.
- **22**: B
- **23**: None of the options match the derived solution.
- **24**: No real solution exists.
- **21**: Cannot determine without additional information. - **22**: B - **23**: No correct option provided. - **24**: No real solution exists.

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