Let's solve question 4 from Part A of the exam:

Question 4

If $2^x = 3$, $3^y = 5$, and $10^z = 8$, express $z$ in terms of $x$ and $y$.

Solution:

We start with the given equations:

1. $2^x = 3$
2. $3^y = 5$
3. $10^z = 8$

Taking the logarithm on both sides of each equation:

1. $x \cdot \log 2 = \log 3$ $\Rightarrow x = \frac{\log 3}{\log 2}$
2. $y \cdot \log 3 = \log 5$ $\Rightarrow y = \frac{\log 5}{\log 3}$
3. $z \cdot \log 10 = \log 8$ $\Rightarrow z = \log 8$ (since $\log 10 = 1$).

Now, $\log 8 = \log(2^3) = 3 \cdot \log 2$. Hence, $z = 3 \cdot \log 2$.

Substitute $\log 2$ in terms of $x$:

From $x = \frac{\log 3}{\log 2}$, we get $\log 2 = \frac{\log 3}{x}$.

Substitute into $z$:

$z = 3 \cdot \frac{\log 3}{x}.$

Expressing $x$ in terms of $y$:

From $y = \frac{\log 5}{\log 3}$, we get $\log 3 = \frac{\log 5}{y}$.

Substitute $\log 3$ into $z$:

$z = 3 \cdot \frac{\frac{\log 5}{y}}{x}.$

Thus, the final expression for $z$ is:

$z = \frac{3 \cdot \log 5}{x \cdot y}.$

This is $z$ expressed in terms of $x$ and $y$.

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Would you like me to simplify this further or solve another question?

Here are related questions for exploration:

1. Can you express $x$ and $y$ independently in terms of $z$?
2. What happens if $z$ has a different base?
3. How would $z$ change if $10^z = 9$?
4. Can you verify these logarithmic transformations numerically?
5. How would you graph these equations?

Tip: Always simplify logarithmic expressions to make substitution easier.