Select the correct answer to the following equation: 27x3/4−1=26

To solve the equation \( 27^{x \frac{3}{4} - 1} = 26 \), we will first isolate \( x \).

1. Rewrite the equation:
\[ 27^{x \frac{3}{4} - 1} = 26 \]

2. Take the logarithm of both sides:
\[ \log(27^{x \frac{3}{4} - 1}) = \log(26) \]

3. Use the power rule of logarithms:
\[ (x \frac{3}{4} - 1) \log(27) = \log(26) \]

4. Solve for \( x \):
\[ x \frac{3}{4} - 1 = \frac{\log(26)}{\log(27)} \]

5. Add 1 to both sides:
\[ x \frac{3}{4} = \frac{\log(26)}{\log(27)} + 1 \]

6. Multiply both sides by \( \frac{4}{3} \):
\[ x = \frac{4}{3} \left( \frac{\log(26)}{\log(27)} + 1 \right) \]

To find an approximate numerical value for \( x \), we can compute the values of \( \log(26) \) and \( \log(27) \) using a calculator:

- \( \log(26) \approx 1.41497 \)
- \( \log(27) \approx 1.43136 \)

Now, substitute these values into the equation:

\[ \frac{\log(26)}{\log(27)} \approx \frac{1.41497}{1.43136} \approx 0.987 \]

So,

\[ x \frac{3}{4} \approx 0.987 + 1 \approx 1.987 \]

Therefore:

\[ x \approx \frac{4}{3} \times 1.987 \approx 2.646 \]

Now we compare the approximate value \( x \approx 2.646 \) with the response choices:

1. \( 1 \)
2. \( 0 \)
3. \( \frac{25}{27} \)
4. \( 3 \)

The closest valid integer response appears to be:

**3.**

So, the answer to the equation is:
**3**