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Chapter 3: Problem 24

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log _{27} 81 $$

Short Answer

Expert verified
The short answer is \(\log_{27} 81 = \frac{4}{3}\).

Step by step solution

01

Convert the given expression to exponent form

The expression we have to evaluate is \(\log_{27} 81\). Now, we need to rewrite this expression in the exponent form. \(a=\log_b c\) can be written as \(b^a=c\), where b is the base, a is the exponent, and c is the result. Therefore, in our case, we need to find the exponent "a" such that: \[27^a = 81\]
02

Express the numbers as powers of 3

To make the equation easier to work with, rewrite the given numbers as powers of 3. 27 can be written as \(3^3\) and 81 can be written as \(3^4\). Replace the numbers in the equation with their corresponding powers of 3. Thus, the equation becomes: \[(3^3)^a = 3^4\]
03

Simplify the equation using exponent laws

According to the power of a power rule, \((a^m)^n = a^{mn}\). Apply this rule to the equation: \[3^{3a} = 3^4\]
04

Solve for "a"

Now, for the equation \(3^{3a} = 3^4\) to be true, the exponents must be equal. Therefore, set the exponents equal to each other: \[3a = 4\] Now, solve for "a": \[a = \frac{4}{3}\]
05

Write the final answer

Since we found that "a" is equal to \(\frac{4}{3}\), the logarithm can be evaluated as: \[\log_{27} 81 = \frac{4}{3}\]

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