Problem 41 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{3} 81 \cdot \log _{2} 16 $$

Step-by-Step Solution

Answer

The expression evaluates to 16.

Step 1: Evaluate \( \log_{3} 81 \)

To evaluate \( \log_{3} 81 \), we need to determine what power 3 must be raised to in order to get 81. Since \( 81 = 3^4 \), we have \( \log_{3} 81 = 4 \).

Step 2: Evaluate \( \log_{2} 16 \)

To find \( \log_{2} 16 \), determine the power to which 2 must be raised to produce 16. Since \( 16 = 2^4 \), it follows that \( \log_{2} 16 = 4 \).

Step 3: Multiply the Results

Now that we have \( \log_{3} 81 = 4 \) and \( \log_{2} 16 = 4 \), multiply these values together: \( 4 \cdot 4 = 16 \).

Key Concepts

Logarithmic Expressions

A logarithmic expression involves a logarithm, which is a mathematical function used to find exponents. In a logarithm, we determine the power to which a specific base must be raised to obtain a given number. The general form of a logarithmic expression is \( \log_b a \), where \( b \) is the base, and \( a \) is the number we want to express as a power of \( b \). Logarithms are particularly helpful because they allow us to express large numbers in a more manageable way.

• For example, \( \log_3 81 \) asks us what power 3 must be raised to achieve 81.
• This expression is the inverse of an exponentiation \( 3^x = 81 \).
• If \( b^x = a \), then \( \log_b a = x \).

Evaluating Logarithms

To evaluate a logarithm, you determine the exponent needed for the base to equal the given number. It involves a bit of detective work. Let's break it down with \( \log_3 81 \) and \( \log_2 16 \).

• Start by asking which exponent will turn the base into the number.
• For \( \log_3 81 \), you find that 3 raised to the 4th power equals 81 because \( 81 = 3^4 \).
• Similarly, for \( \log_2 16 \), you determine that 2 raised to the 4th power equals 16 because \( 16 = 2^4 \).

Remember: The key is identifying the exponent needed, making logarithms a powerful tool for working backward from exponentiation.

Exponents

Exponents, also known as powers, are a shorthand way to express repeated multiplication of a number by itself. In the expression \( b^x \), \( b \) is the base and \( x \) is the exponent, which tells us how many times to multiply \( b \) by itself. For example, \( 3^4 \) means multiplying four 3s together: \( 3 \times 3 \times 3 \times 3 = 81 \).

• Exponents are essential in simplifying and solving logarithmic expressions.
• They help find the inverse operation in logarithms, such as determining the number for \( \log_b a \).
• Understanding the relationship between exponents and logarithms is crucial for evaluating log expressions accurately.

Base Conversions

Base conversions are essential when dealing with logarithms, especially when the logarithm bases differ. Although you may work with different bases when evaluating logarithms, understanding how to convert between bases can simplify the process.

• The change of base formula is useful: \( \log_b a = \frac{\log_c a}{\log_c b} \) where \( c \) is a new base, often set to 10 or \( e \) (the natural logarithm).
• This formula allows you to convert logarithms from one base to another, especially when using a calculator that may only support base 10 or \( e \).
• Knowing base conversions is also useful in practical applications where different base systems are used, such as binary systems (base 2) in computing.