Which exponential equation is equivalent to this logarithmic equation?
$\log x=4$
The problem asks for the exponential form equivalent to the logarithmic equation \(\log x = 4\). In standard mathematical notation, the expression \(\log x\) without an explicit base denotes the common logarithm, which has base 10. Thus, the equation can be written as \(\log_{10} x = 4\).
The fundamental definition of a logarithm states that for any base \(b > 0\) (where \(b eq 1\)), the logarithmic equation \(\log_b(y) = x\) is equivalent to the exponential equation \(b^x = y\). Here, our equation is \(\log_{10} x = 4\). Identifying the components: the base \(b\) is \(10\), the argument \(y\) is \(x\), and the exponent \(x\) in the definition corresponds to \(4\) in our equation.
Using the equivalence \(b^c = a\) where \(\log_b(a) = c\), we substitute the identified values: \(b = 10\), \(c = 4\), and \(a = x\). This gives us the exponential form: \(10^4 = x\). Comparing this to the given options, it matches the second option.
Understanding how to convert logarithmic expressions to their exponential counterparts is essential in fields like finance and science. For instance, the Richter scale used to measure earthquakes is a logarithmic scale, where each whole number increase in magnitude corresponds to a tenfold increase in the measured amplitude. Converting between logarithmic and exponential forms allows scientists to determine the actual ground movement (amplitude) associated with a given magnitude, helping in building safer infrastructure in earthquake-prone regions.