Which exponential equation is equivalent to this logarithmic equation?
$\log x=4$
We are given the logarithmic equation \(\log x = 4\). The goal is to identify its equivalent exponential form from the provided options. By convention, in mathematics, when a base is not specified for a logarithm, the notation \(\log x\) refers to the common logarithm, which has a base of 10. Therefore, \(\log x\) is equivalent to \(\log_{10} x\).
The fundamental definition of a logarithm states that for any base \(b > 0\) (where \(b eq 1\)), the equation \(\log_b A = C\) is equivalent to the exponential equation \(b^C = A\). In our case, the base \(b\) is 10, the argument \(A\) is \(x\), and the exponent \(C\) is 4. Substituting these values into the exponential form \(b^C = A\), we get \(10^4 = x\).
Comparing our derived result \(10^4 = x\) with the given options:
We can see that the correct choice is \(10^4 = x\).
Logarithms are essential in measuring scales that span huge ranges of values, such as the pH scale for acidity or the Richter scale for earthquake magnitude. For example, if you are measuring the pH of a solution where the concentration of hydrogen ions is \(x\), the relationship is defined by \(\log(x) = -pH\). Knowing how to convert between logarithmic and exponential forms allows scientists to solve for the concentration of substances directly from the pH value, making it a powerful tool in chemistry and environmental science.