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Chapter 3: Problem 121

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

Short Answer

Expert verified
\(\log_3 81 = 4\)

Step by step solution

01

Definition of Logarithms

Logarithms are the opposites of exponentials. In the context of an exponential equation \(3^n = 81\), the logarithm base 3 of 81, denoted by \(\log _{3} 81 ?\), solves for the exponent \(n\). Therefore, the expression \(\log _{3} 81\) asks: '3 should be raised to which power to get 81?'.
02

Properties of a Logarithm Base

In a logarithmic function \(\log_b a = n\), the base \(b\) raised to the power \(n\) equals \(a\). Here, the base is 3. So, to evaluate \(\log _{3} 81\), consider the powers of 3 until 81 is reached.
03

Calculate Logarithm

Calculate \(\log_3 81\) by finding \(n\) in the equation \(3^n = 81\). By checking the powers of 3: \(3^1 = 3\), \(3^2 = 9\), \(3^3 = 27\), and \(3^4 = 81\). Hence, \(3^4 = 81\), so the logarithm base 3 of 81 equals 4. That is \(\log_3 81 = 4\).

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