Let's analyze and solve the expression shown in the image.

The given expression is:

$\frac{{^3\log 243 + ^2\log 32 - ^3\log 9}}{{^3\log 27 + ^2\log^2 8}}$

Where:

• $^n\log a$ means $\log(a)$ to the base $n$, or $\log_n(a)$.

Steps for Simplification:

1. Simplify $^3\log 243$:

Using the logarithm change of base rule, we know $^n\log a = \frac{\log(a)}{\log(n)}$, and: 243 = 3^5 \quad \Rightarrow \quad ^3\log 243 = 5

2. Simplify $^2\log 32$:

32 = 2^5 \quad \Rightarrow \quad ^2\log 32 = 5

3. Simplify $^3\log 9$:

9 = 3^2 \quad \Rightarrow \quad ^3\log 9 = 2

4. Simplify $^3\log 27$:

27 = 3^3 \quad \Rightarrow \quad ^3\log 27 = 3

5. Simplify $^2\log^2 8$:

Since $\log^2 8$ means $(\log 8)^2$, let's first simplify: 8 = 2^3 \quad \Rightarrow \quad ^2\log 8 = 3 Thus: $^2\log^2 8 = 3^2 = 9$

Now, substitute these values into the expression:

$\frac{{5 + 5 - 2}}{{3 + 9}} = \frac{{8}}{{12}} = \frac{2}{3}$

The value of the entire expression is $\frac{2}{3}$.

Would you like any further details or clarifications on this solution?

Here are 5 related questions to extend your understanding:

1. How do you apply the change of base rule for logarithms?
2. What happens when the base and the argument of a logarithm are powers of the same number?
3. How would the expression change if we altered the base of the logarithms?
4. What are some common properties of logarithms that simplify expressions?
5. How does logarithmic squaring work in more complex expressions?

Tip:

When simplifying logarithms, always check if the argument is a power of the base, as this allows for quick simplifications!