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Chapter 3: Problem 107

Evaluate each expression without using a calculator. $$\log _{2}\left(\log _{3} 81\right)$$

Short Answer

Expert verified
The value of the expression \( \log_{2}(\log_{3} 81) \) is 2.

Step by step solution

01

Evaluate the inner logarithm

Start with the inner log first. The expression is \( \log_{3} 81 \). We know that \(3^{4}\) equals to \(81\), so that means \( \log_{3} 81 = 4 \). So the original equation simplifies to \( \log_{2}(4) \).
02

Evaluate the outer logarithm

Next, we evaluate the outer logarithm. The expression is now \( \log_{2} 4 \). Here, we know that \(2^2\) equals to \(4\), so that means \( \log_{2} 4 = 2 \). This is the value of the original expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a mathematical shorthand used to express repeated multiplication. When we see a number with an exponent, like \(3^4\), it means multiplying 3 by itself four times (\(3 \times 3 \times 3 \times 3\)). This operation results in 81.
The exponent indicates how many times the base, in this case 3, is used as a factor.
Some key properties of exponents are:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{mn}\)
  • Power of a Product: \((ab)^n = a^n \cdot b^n\)
Understanding these properties can help simplify expressions and solve equations with exponents more easily.
Base Conversion
In the context of logarithms, base conversion refers to understanding the relationship between different bases. Here we're dealing with \( \log_{3}\left(81\right) \) and \( \log_{2}\left(4\right) \).
Let's take a closer look:
  • Base-3 Logarithm: We're looking for what power we need to raise 3 to get 81. Since \(3^4 = 81\), we find that \( \log_{3}81 = 4\).
  • Base-2 Logarithm: For \( \log_{2}\left(4\right)\), we need to determine the power to which 2 must be raised to yield 4. Since \(2^2 = 4\), we conclude that \( \log_{2}4 = 2\).
Converting bases helps to simplify complex logarithmic operations by enabling us to interpret expressions more directly.
Evaluate Expressions
Evaluating expressions in mathematics involves finding the value of an expression. In the original exercise, we are tasked with evaluating \( \log _{2}\left( \log _{3} 81\right) \) without using a calculator.
Let's break it down step-by-step:
  • Inner Logarithm: Start by evaluating the inner part of the expression, \( \log_{3} 81\). Since we know \(3^4 = 81\), this becomes 4.
  • Outer Logarithm: With the inner part evaluated to 4, our expression now becomes \( \log_{2} 4\). Since \(2^2 = 4\), this simplifies to 2.
The final answer to the expression is 2, achieved through understanding exponents and the properties of logarithms. Each step logically follows the previous, illustrating the power of simplifying expressions by breaking them into smaller, more manageable parts.

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$$\text { Solve for } y: 7 x+3 y=18$$
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