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The problem states that \( t \) is a positive real number, and you are given the equation:

\[
\log_3 t + \log_9 t + \log_{27} t + \log_{81} t = 10
\]

We are asked to find the value of \( t \) and express the answer in the form \( 3^r \), where \( r \in \mathbb{Q} \) (the set of rational numbers). The hint provided is to use the formula:

\[
\log_a b = \frac{\log_c b}{\log_c a}
\]

### Step-by-step solution:

#### 1. Express each logarithm with base 3
We will rewrite each term using base 3 logarithms, using the change of base formula: 

\[
\log_a b = \frac{\log_c b}{\log_c a}
\]

- \( \log_9 t = \log_{3^2} t = \frac{1}{2} \log_3 t \)
- \( \log_{27} t = \log_{3^3} t = \frac{1}{3} \log_3 t \)
- \( \log_{81} t = \log_{3^4} t = \frac{1}{4} \log_3 t \)

#### 2. Substitute back into the original equation
Now, substitute these expressions back into the given equation:

\[
\log_3 t + \frac{1}{2} \log_3 t + \frac{1}{3} \log_3 t + \frac{1}{4} \log_3 t = 10
\]

Let \( x = \log_3 t \). The equation becomes:

\[
x + \frac{1}{2}x + \frac{1}{3}x + \frac{1}{4}x = 10
\]

#### 3. Combine like terms
We can factor out \( x \):

\[
x \left( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right) = 10
\]

Now, calculate the sum inside the parentheses:

\[
1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{12}{12} + \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{25}{12}
\]

So, the equation becomes:

\[
x \times \frac{25}{12} = 10
\]

#### 4. Solve for \( x \)
Multiply both sides of the equation by \( \frac{12}{25} \):

\[
x = 10 \times \frac{12}{25} = \frac{120}{25} = 4.8
\]

Thus, \( \log_3 t = 4.8 \).

#### 5. Solve for \( t \)
Now, using the definition of logarithms:

\[
t = 3^{4.8}
\]

Thus, the value of \( t \) is \( 3^{4.8} \).

---

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

Here are some related questions for further exploration:

1. How can the change of base formula be used to express logarithms in terms of natural logarithms?
2. What is the relationship between rational exponents and logarithms?
3. How does the sum of fractions play a role in solving logarithmic equations like this?
4. Can this method be generalized to work with other bases, such as base 2?
5. What is the value of \( t \) in decimal form if expressed without logarithmic notation?

**Tip**: When combining logarithms with different bases, always try to express them with a common base to simplify the equation.

t is a positive real number, with log₃(t) + log₉(t) + log₂₇(t) + log₈₁(t) = 10. Find the value of t. Give your answer in the form 3^r, where r ∈ ℚ.

Solve the equation log₃(t) + log₉(t) + log₂₇(t) + log₈₁(t) = 10. This problem involves logarithmic properties and the change of base formula. Learn how to express t in the form 3^r with step-by-step instructions.

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