/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Evaluate each logarithm. $$ ... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 85

Evaluate each logarithm. $$ \log _{36} 6 $$

Short Answer

Expert verified
The evaluated logarithm is \(1/2\).

Step by step solution

01

Understand the properties of Logarithms

A logarithm \(\log_b a\) can be understood as the power (exponent) to which the base (\(b\)) must be raised in order to obtain \(a\). So in \(\log _{36} 6\), we need to determine what power 36 needs to be raised to, in order to get 6.
02

Simplify the Base

Given our base is 36, we rewrite the base (36) as \(6^2\). This allows us to see how our base and the number we want it to transform into (6) are related.
03

Apply Logarithm Rules

Applying logarithm rules, specifically the power rule which states that \(log_b{a^n} = n \cdot log_b{a}\), we can write \(\log _{36} 6\) as \(\log _{6^2} 6\), which simplifies to \(1/2 \cdot \log _{6} 6\). This follows as we have taken the 2 from the base and brought it in front of the log.
04

Evaluate the Logarithm

When the base and the argument of the log are the same number (\(log_b b\)), the log equals 1. Therefore, \(log _{6} 6\) simplifies to 1, leaving us with \(1/2 \cdot 1 = 1/2 \) as the final solution.

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