What is the solution of $\log _x 729=3$ ?
We are given the logarithmic equation \(\log_x 729 = 3\), where \(x\) is the base of the logarithm. By the definition of a logarithm, the expression \(\log_b a = c\) is equivalent to the exponential form \(b^c = a\). Here, the base is \(x\), the exponent is \(3\), and the result is \(729\). Thus, we can rewrite the equation as \(x^3 = 729\). Note that for a logarithm \(\log_x 729\) to be defined, the base \(x\) must be a positive real number and \(x eq 1\).
To solve for \(x\), we need to find the cube root of \(729\). We are looking for a value \(x\) such that \(x \times x \times x = 729\). Testing the given options:
This value satisfies the conditions \(x > 0\) and \(x eq 1\), confirming that \(x = 9\) is the correct base for the logarithm.
Having determined that \(x = 9\), we can see that this corresponds to option (a). The steps followed ensure that the definition of the logarithm is respected and the exponential equation is correctly solved. Thus, the solution to the equation \(\log_x 729 = 3\) is \(9\).
Logarithms are essential in measuring quantities that change exponentially, such as calculating the magnitude of earthquakes using the Richter scale or determining the pH level of chemical solutions. In this problem, finding the base of a logarithm demonstrates how we can reverse exponential growth to understand the starting factor, a fundamental skill in fields like finance for calculating interest rates or in biology for modeling population growth. Understanding this relationship helps in solving problems where you need to identify the base rate of change given a final result.