To solve the equation [tex]\(\log _x 729 = 3\)[/tex], we need to understand and manipulate logarithms. Here's a step-by-step solution:
1. Start with the given equation:
[tex]\[
\log_x 729 = 3
\][/tex]
2. Recall that [tex]\(\log_x 729 = 3\)[/tex] means that [tex]\(x\)[/tex] raised to the power of 3 equals 729. In exponential form, this is:
[tex]\[
x^3 = 729
\][/tex]
3. To solve for [tex]\(x\)[/tex], we need to find the cube root of 729:
[tex]\[
x = \sqrt[3]{729}
\][/tex]
4. The cube root of 729 is 9 because:
[tex]\[
9^3 = 9 \times 9 \times 9 = 729
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{9}
\][/tex]
