Which logarithmic equation is equivalent to $3^2=9$ ?
The exponential equation is given as \(3^2 = 9\). In an exponential expression of the form \(b^e = v\), the variable \(b\) represents the base, \(e\) is the exponent, and \(v\) is the result. For the equation \(3^2 = 9\), we have the base \(b = 3\), the exponent \(e = 2\), and the value \(v = 9\). The objective is to convert this expression into its equivalent logarithmic form.
The definition of a logarithm provides a direct way to convert between exponential and logarithmic notation. Specifically, for a base \(b > 0\) (where \(b eq 1\)), the equation \(b^e = v\) is equivalent to the logarithmic equation \(\log_b v = e\). This means the logarithm \(\log_b v\) answers the question: "To what power must the base \(b\) be raised to produce the value \(v\)?" In our case, the base \(b = 3\) must be raised to the power of \(2\) to equal \(9\).
By substituting our values \(b = 3\), \(v = 9\), and \(e = 2\) into the definition \(\log_b v = e\), we obtain \(\log_3 9 = 2\). This equation is equivalent to \(2 = \log_3 9\). We can now evaluate the given options:
The first option, \(2 = \log_3 9\), matches our derived expression perfectly.
Logarithms are used extensively in fields where values span many orders of magnitude, such as measuring the intensity of sounds (decibels) or earthquakes (Richter scale). For example, if you are calculating the pH of a solution, which is defined as \(pH = -\log_{10}[H^+]\), you are essentially reversing the process of exponentiation to find the concentration of hydrogen ions \([H^+]\). Understanding this relationship allows scientists to work with very small or very large numbers more conveniently.