To solve the expression:

$³\log 81 + ⁴\log 32 / ⅑\log 27$

we need to break it down and simplify each term.

Step 1: Simplifying each logarithmic term

1. $³\log 81$ represents the logarithm of 81 in base 3. We know that $81 = 3^4$, so:

   $³\log 81 = 4$

2. $⁴\log 32$ represents the logarithm of 32 in base 4. We know that $32 = 2^5$, and this doesn't simplify nicely in base 4. So we can use the change of base formula:

   $⁴\log 32 = \frac{\log 32}{\log 4}$

   Using logarithmic properties:

   $\log 32 = 5\log 2, \quad \log 4 = 2\log 2$

   Thus:

   $⁴\log 32 = \frac{5\log 2}{2\log 2} = \frac{5}{2}$

3. $⅑\log 27$ represents the logarithm of 27 in base 9. We know that $27 = 3^3$, and using the change of base formula again:

   $⅑\log 27 = \frac{\log 27}{\log 9}$

   $\log 27 = 3\log 3$ and $\log 9 = 2\log 3$, so:

   $⅑\log 27 = \frac{3\log 3}{2\log 3} = \frac{3}{2}$

Step 2: Plugging the values back into the original expression

Now, substitute the values into the expression:

$³\log 81 + ⁴\log 32 / ⅑\log 27 = 4 + \frac{\frac{5}{2}}{\frac{3}{2}}$

The division simplifies as follows:

$\frac{\frac{5}{2}}{\frac{3}{2}} = \frac{5}{2} \times \frac{2}{3} = \frac{5}{3}$

Thus the entire expression becomes:

$4 + \frac{5}{3}$

Step 3: Simplifying the final expression

Convert 4 into a fraction with denominator 3:

$4 = \frac{12}{3}$

Now add:

$\frac{12}{3} + \frac{5}{3} = \frac{17}{3}$

Final Answer:

$³\log 81 + ⁴\log 32 / ⅑\log 27 = \frac{17}{3}$

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do you change the base of a logarithm?
2. What is the change of base formula and when is it used?
3. How do you simplify complex fractions?
4. Can you explain how logarithmic properties help simplify log equations?
5. How can you solve logarithmic equations with different bases?

Tip: When dealing with logarithms of different bases, use the change of base formula to make calculations easier!