24. The truth set of [tex]3^{x-7} \times 3^x = 27[/tex] is:
- a) 4
- b) 5
- c) 2

25. The truth set of [tex]4(5^x) = 5^{2x} - 5[/tex] is:
- a) [tex]\(\{-1\}\)[/tex]
- b) [tex]\(\{0, 1\}\)[/tex]

26. Which of the following is not true?
- a) If [tex]\(a \ \textgreater \ 1, f(x) = a^x\)[/tex] is an increasing function.
- b) If [tex]\(a \ \textless \ 1, f(x) = a^x\)[/tex] is a decreasing function.
- c) If [tex]\(0 \ \textless \ a \ \textless \ 1\)[/tex], then [tex]\(a^x \ \textless \ 1\)[/tex] for [tex]\(x \ \textless \ 0\)[/tex].
- d) For [tex]\(a \ \textgreater \ b \ \textgreater \ 1\)[/tex], if [tex]\(a^x = b^x\)[/tex], then [tex]\(x = 0\)[/tex].

27. If [tex]\(\frac{5^x}{25^{x-1}} = 125^x\)[/tex], then what is the value of [tex]\(x\)[/tex]?
- a) [tex]\(\frac{1}{2}\)[/tex]
- b) [tex]\(\frac{1}{4}\)[/tex]
- c) [tex]\(\frac{-1}{4}\)[/tex]
- d) [tex]\(-\frac{1}{2}\)[/tex]

28. What is the factor of [tex]\(x^5 - x^4 - 6x^3\)[/tex]?
- a) [tex]\(x^2(x^2 - x - 6)\)[/tex]
- b) [tex]\(x^3(x^2 + x - 6)\)[/tex]
- c) [tex]\(x^3(x + 2)(x - 3)\)[/tex]
- d) [tex]\(x^3(x + 2)(x + 3)\)[/tex]

29. If [tex]\(\log_3(x^2 - 54) = 3\)[/tex], then [tex]\(x\)[/tex] is equal to:
- a) 9
- b) -9
- c) a and b
- d) 2

Part III: Answer the following questions with the necessary steps.

1. Find the exact value of
[tex]\[
(\log_4 3)(\log_5 4)(\log_6 5)(\log_7 6)(\log_8 7)(\log_9 8)
\][/tex]
(2 points)

2. Solve each of the following:
- a) [tex]\(4^{2x} = 8^{3x + 1}\)[/tex]
- b) [tex]\(4^x \left(\frac{1}{2}\right)^{3 - 2x} = 8 \cdot (2^x)^2\)[/tex]
- c) [tex]\(3^{x - 1} \times 3^{1 - x} = 3\)[/tex]



Answer :

## Question 24: Solve [tex]\( 3^{x-7} \times 3^x = 27 \)[/tex]

### Step-by-Step Solution:

1. Combine the exponential expressions:
[tex]\[ 3^{x-7} \times 3^x = 27 \][/tex]
Using the property of exponents [tex]\(a^m \times a^n = a^{m+n}\)[/tex], we combine the exponents:
[tex]\[ 3^{(x-7) + x} = 27 \][/tex]
Simplify the exponent:
[tex]\[ 3^{2x-7} = 27 \][/tex]

2. Rewrite 27 as a power of 3:
We know that [tex]\(27 = 3^3\)[/tex], thus:
[tex]\[ 3^{2x-7} = 3^3 \][/tex]

3. Equate the exponents:
Since the bases are the same and the equation [tex]\(a^m = a^n\)[/tex] implies [tex]\(m=n\)[/tex], we equate the exponents:
[tex]\[ 2x - 7 = 3 \][/tex]

4. Solve for [tex]\(x\)[/tex]:
Solve the linear equation for [tex]\(x\)[/tex]:
\begin{align}
2x - 7 &= 3 \\
2x &= 3 + 7 \\
2x &= 10 \\
x &= \frac{10}{2} \\
x &= 5
\end{align
}

Thus, the truth set of [tex]\( 3^{x-7} \times 3^x = 27 \)[/tex] is [tex]\( \boxed{5} \)[/tex].

Let's verify the intermediate steps:
- The left-hand side simplification is [tex]\(2x - 7\)[/tex],
- We identified [tex]\(27\)[/tex] as [tex]\(3^3\)[/tex],
- This led to the equation [tex]\(2x - 7 = 3\)[/tex], and solving for [tex]\(x\)[/tex] yielded [tex]\(x = 5\)[/tex].

Given these verified intermediate results, [tex]\(x = 5\)[/tex] is indeed the correct solution.

So, [tex]\( c) 5 \)[/tex] is the correct answer.