Which exponential equation is equivalent to this logarithmic equation?
$\log _2 x=24$
The given equation is a logarithmic equation in the form \(\log_b(a) = c\). To determine the equivalent exponential form, we need to identify the base (\(b\)), the argument (\(a\)), and the exponent (\(c\)) from the given equation \(\log_2 x = 24\). Here, the base \(b = 2\), the argument \(a = x\), and the exponent \(c = 24\).
The fundamental definition of a logarithm states that the expression \(\log_b(a) = c\) is equivalent to the exponential expression \(b^c = a\). This means that raising the base \(b\) to the power of \(c\) yields the argument \(a\).
By substituting the values we identified in step 1 (\(b=2\), \(a=x\), \(c=24\)) into the definition \(b^c = a\), we get \(2^{24} = x\). Comparing this result with the provided options, we can see that it corresponds to the second option.
Logarithms are essential in measuring things that change at extremely different scales. For instance, the Richter scale used for earthquakes or the pH scale in chemistry are logarithmic. If you know the earthquake's magnitude (the logarithm of its intensity), you can easily calculate its actual intensity using the inverse operation, exponentiation. Similarly, this helps in understanding data processing, acoustics (decibels), and even compound interest calculations in finance. Understanding the relationship between logs and exponents is the key to unlocking these powerful tools.