Problem 89 - College Algebra

Question

What question can be asked to help evaluate \(\log _{3} 81 ?\)

Step-by-Step Solution

Answer

The log base 3 of 81 is 4. This can be deduced by re-writing the logarithm function in exponential form and comparing the exponents.

Step 1: Understand the Logarithm

The logarithm function \(\log _{a} b = n\) is equivalent to the expression \(a^n = b\). This means that \(n\) is the power to which \(a\) must be raised to get \(b\). In the provided problem, \(a = 3\) and \(b = 81\). We are to find \(n\). The aim is to evaluate \(\log _{3} 81\).

Step 2: Rewrite in Exponential Form

Rewrite the logarithmic function in exponential form: \(3^n = 81\). We know that \(81\) is the same as \(3^4\) because \(3*3*3*3 = 81\). So we have \(3^n = 3^4\).

Step 3: Compare the Exponents

The bases on both sides of the equation are equal (\(3 = 3\)), so the exponents should also be equal. That means \(n = 4\). This is possible due to the property that if \(a^b = a^c\) then \(b = c\) if \(a > 0\) and \(a ≠ 1\).