What is the solution of $\log _x 729=3$ ?
We need to solve the logarithmic equation \(\log_x 729 = 3\) for the base \(x\). The fundamental definition of a logarithm states that for a base \(b > 0\) and \(b eq 1\), the equation \(\log_b A = C\) is equivalent to the exponential form \(b^C = A\). In this problem, the base is \(x\), the exponent is 3, and the result is 729. Therefore, we can rewrite the equation as \(x^3 = 729\).
To find the value of \(x\), we need to determine which number, when raised to the power of 3, equals 729. This is equivalent to calculating the cube root of 729, which is \(x = \sqrt[3]{729}\). We can test values to solve this. Since \(9 \times 9 = 81\) and \(81 \times 9 = 729\), it follows that \(9^3 = 729\). Thus, \(x = 9\).
We have found that \(x = 9\). Comparing this result with the provided options:
The solution matches option a. Thus, the correct value for \(x\) is \(\boxed{9}\).
Logarithms are widely used in various real-world applications, such as calculating the magnitude of earthquakes using the Richter scale or measuring sound intensity in decibels. In finance, they help in determining the time required for an investment to grow to a certain amount when compounded periodically. Solving equations like \(\log_x 729 = 3\) helps understand the relationship between exponential growth and logarithmic decay, providing a foundation for analyzing complex data in science and engineering.