Sure, I'll assist you with both parts of the question step by step.
### Part (i)
We need to find the value of [tex]\( x \)[/tex] for the equation:
[tex]\[
\log_2(2x) = \log_2(5x + 4) - 3
\][/tex]
1. First, let's isolate the logarithmic terms on one side:
[tex]\[
\log_2(2x) + 3 = \log_2(5x + 4)
\][/tex]
2. Recognize that adding 3 to the logarithm is equivalent to multiplying by [tex]\( 2^3 = 8 \)[/tex] inside the logarithm:
[tex]\[
\log_2(2x) + 3 = \log_2(8 \cdot 2x)
\][/tex]
3. Simplify inside the logarithm:
[tex]\[
\log_2(8 \cdot 2x) = \log_2(16x)
\][/tex]
4. Now we equate the two logarithms:
[tex]\[
\log_2(16x) = \log_2(5x + 4)
\][/tex]
5. Since the logarithms are equal, their arguments must be equal as well:
[tex]\[
16x = 5x + 4
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
16x - 5x = 4
\][/tex]
[tex]\[
11x = 4
\][/tex]
[tex]\[
x = \frac{4}{11}
\][/tex]
So the exact value of [tex]\( x \)[/tex] is [tex]\( \frac{4}{11} \)[/tex].
### Part (ii)
We need to express [tex]\( y \)[/tex] in terms of [tex]\( a \)[/tex] given the equation:
[tex]\[
\log_a y + 3 \log_a 2 = 5
\][/tex]
1. Utilize the logarithmic property [tex]\( \log_b(m^n) = n \log_b m \)[/tex]:
[tex]\[
\log_a y + \log_a (2^3) = 5
\][/tex]
2. Simplify [tex]\( 2^3 \)[/tex]:
[tex]\[
\log_a y + \log_a 8 = 5
\][/tex]
3. Combine the logarithms:
[tex]\[
\log_a (y \cdot 8) = 5
\][/tex]
4. Express the logarithmic equation in exponential form:
[tex]\[
y \cdot 8 = a^5
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{a^5}{8}
\][/tex]
So [tex]\( y \)[/tex] in its simplest form is [tex]\( \frac{a^5}{8} \)[/tex].
### Summary
- For part (i), the exact value of [tex]\( x \)[/tex] is [tex]\( \frac{4}{11} \)[/tex].
- For part (ii), [tex]\( y \)[/tex] expressed in terms of [tex]\( a \)[/tex] is [tex]\( \frac{a^5}{8} \)[/tex].
