Problem 139 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

Step-by-Step Solution

Answer

The statement makes sense. The exponential equation \(4^{x} = 15\) can indeed be written, and thus solved, in its equivalent logarithmic form as \(\log_{4}(15) = x\).

Step 1: Understanding the Exponential and Logarithmic form

An equation is in exponential form if it is written as '\(b^{a} = c\)', where \(b\) is the base, \(a\) is the exponent and \(c\) is the result. It can be converted into logarithmic form: \(\log_{b}(c) = a\). These two are equivalent.

Step 2: Converting the given equation into Logarithmic Form

The given equation is \(4^{x}=15\). Convert that into logarithmic form gives, \(\log_{4}(15) = x\).

Step 3: Explanation of the Conversion

In the transformation, the base of the exponent becomes the base of the logarithm (\(4\) in this case), the result or output of the exponential expression is the input or argument of the logarithmic function (which is \(15\)), and the exponents are identical in both forms (\(x\) in this case).