Problem 107
Question
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{2}\left(\log _{3} 81\right) $$
Step-by-Step Solution
Verified Answer
The value of the given expression is 2.
1Step 1: Simplifying Inner Logarithm
The given problem has nested logarithms. To simplify it, start from the innermost logarithm and work outward. The innermost log here is \( \log _{3} 81 \). 81 is \( 3^4 \), so \( \log _{3} 81 = 4 \)
2Step 2: Substitute and Solve
Next substitute the value of \( \log _{3} 81 \) into the original problem. The expression becomes \( \log _{2} 4 \). In other words, we're asking to what power should we raise 2 to get 4. The answer to this is 2 because \( 2^2 = 4 \). So, \( \log _{2} 4 = 2 \)
Key Concepts
Understanding Nested LogarithmsEvaluating Logarithmic ExpressionsMastering Logarithm Properties
Understanding Nested Logarithms
Nested logarithms are simply logarithmic expressions that appear within other logarithms. These expressions require you to solve from the innermost log to the outermost log. The key to solving nested logarithms is to first evaluate the inner logarithm, just like peeling an onion layer by layer.
For instance, if you have something like \( \log_{2}(\log_{3} 81) \), the first step is to simplify \( \log_{3} 81 \). This means determining what power you need to raise 3 to get 81. Since \( 81 = 3^4 \), \( \log_{3} 81 = 4 \).
Once the innermost logarithm is evaluated, you can then substitute back into the expression and focus on the next layer, which is \( \log_{2} 4 \). Solving nested logarithms can seem tricky, but remember: step by step, from the inside out!
For instance, if you have something like \( \log_{2}(\log_{3} 81) \), the first step is to simplify \( \log_{3} 81 \). This means determining what power you need to raise 3 to get 81. Since \( 81 = 3^4 \), \( \log_{3} 81 = 4 \).
Once the innermost logarithm is evaluated, you can then substitute back into the expression and focus on the next layer, which is \( \log_{2} 4 \). Solving nested logarithms can seem tricky, but remember: step by step, from the inside out!
Evaluating Logarithmic Expressions
When evaluating a logarithmic expression, you need to identify what the logarithm is asking. A logarithm \( \log_b x \) is essentially asking: "To what power must \( b \) be raised to achieve \( x \)?".
Evaluating such expressions becomes intuitive once you grasp that all you're doing is finding powers. Think of it as a reflection, where you look backward from the result (81) to the power (4) that creates it using base (3).
- Start by identifying the base \( b \).
- Find out the number (the exponent) which gives you \( x \) when the base is raised to it.
Evaluating such expressions becomes intuitive once you grasp that all you're doing is finding powers. Think of it as a reflection, where you look backward from the result (81) to the power (4) that creates it using base (3).
Mastering Logarithm Properties
Logarithm properties are invaluable tools that help simplify and solve logarithmic expressions. They work much like rules of operations for numbers and include the following:
- Power Rule: \( \log_b(x^y) = y \cdot \log_b(x) \).
- Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \).
- Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).
- Change of Base Formula: Allows evaluating logs with unfamiliar bases using \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), with \( k \) as any positive number.
Other exercises in this chapter
Problem 106
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{5}\left(\log _{2} 32\right) $$
View solution Problem 107
Describe the power rule for logarithms and give an example.
View solution Problem 108
Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)
View solution Problem 108
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log (\ln e) $$
View solution