Problem 89 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Evaluate the expression in part (a) without using a calculator. b. Use your result from part (a) to write the expression in part (b) as a single logarithm whose coefficient is 1 a. \(\log _{3} 9\) b. \(\log _{3} x+4 \log _{3} y-2\)

Step-by-Step Solution

Answer

a. The value of \(\log_{3}9\) is 2. b. The expression \(\log_{3}x + 4\log_{3}y - 2\) simplifies to \(\log_{3}\left(\frac{xy^4}{9}\right)\).

Step 1: Evaluate the expression \(\log _{3}9\)

Remember that logarithm is the inverse operation to exponentiation. The expression \(\log_{3}9\) means which exponent on base 3 gives the number 9. Since 3 squared (3^2) is 9, the value of \(\log_{3}9\) is 2.

Step 2: Start the simplification of the second expression

The second expression given is \(\log_{3}x + 4\log_{3}y - 2\). Start by observing the term '4\log_{3}y'. Multiplying a logarithm by a number means taking the number to the power of the logarithmic result, according to the log power rule. Consequentially, 4\log_{3}y is equal to \(\log_{3}(y^4)\). The expression can be rewritten as \(\log_{3}x + \log_{3}(y^4) - 2\) to simplify the process.

Step 3: Use the rule of logs to simplify the first two terms

The sum of two logarithms with the same base can be written as a single logarithm with the resulting logarithmic result being the product of the logarithms. This can be restated as log_b(M) + log_b(N) = log_b(MN). Thus, \(\log_{3}x + \log_{3}(y^4)\) simplifies to \(\log_{3}(xy^4)\). This gives an updated expression of \(\log_{3}(xy^4)-2\).

Step 4: Simplify the expression to a single logarithm

The final step is to convert the remaining subtraction into a division inside a single logarithm. From step 1, we know that 2 is equivalent to \(\log_{3}9\). Thus, subtracting 2 means dividing by 9, according to the log quotient rule. Therefore, \(\log_{3}(xy^4) - 2\) is simplified to \(\log_{3}\left(\frac{xy^4}{9}\right)\). This is the final result.

Key Concepts

Logarithm Exponentiation Relationship

Understanding the relationship between logarithms and exponentiation is vital for simplifying logarithmic expressions effectively. A logarithm answers the question: 'To what power do we need to raise a specific base to obtain a certain number?' For instance, when we see \( \log_{3}9 \), we're essentially asking, '3 raised to what power gives us 9?' Since \( 3^2 = 9 \), the answer is 2, which is why \( \log_{3}9 = 2 \).

This interplay between logarithms and exponents is a foundational concept that helps students manipulate and combine logarithmic terms, leading to simplification of complex expressions. It's like a language translation between two mathematical 'dialects': the world of multiplication and powers (exponentiation) on one side and the realm of logarithms on the other.

Logarithm Properties

Logarithms follow specific rules, or properties, that allow us to perform operations on them just like we would with regular numbers. Some of the most important ones include:

• Product Rule: \( \log_b(M) + \log_b(N) = \log_b(MN) \), which tells us that the sum of logarithms is the log of the product of their arguments.
• Quotient Rule: \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \), indicating that the difference between two logs is the log of the quotient of their arguments.
• Power Rule: \( n\log_b(M) = \log_b(M^n) \), meaning that a coefficient in front of a log can be moved to become an exponent within the log.

By applying these properties, we can turn a complex expression involving multiple logarithmic terms into a much simpler form. Not only does this make the expressions easier to work with, but it also prepares them to be solved or further manipulated as needed.

Simplifying Logarithmic Expressions

Simplification of logarithmic expressions leverages the logarithm properties we've discussed. Take the expression \( \log_{3}x + 4\log_{3}y - 2 \). To transform it into a single logarithm, first, apply the power rule to rewrite \( 4\log_{3}y \) as \( \log_{3}(y^4) \). Now, we have \( \log_{3}x + \log_{3}(y^4) \).

Next, apply the product rule to combine \( \log_{3}x \) and \( \log_{3}(y^4) \) into \( \log_{3}(xy^4) \). Lastly, we tackle the subtraction of 2 by invoking the quotient rule. Remembering that 2 is equivalent to \( \log_{3}9 \), the expression simplifies to \( \log_{3}\left(\frac{xy^4}{9}\right) \).

The key to mastering these simplifications is to practice the proper application of logarithm properties in sequence to reduce and consolidate logarithmic terms, as displayed in the step by step solution provided. A solid grasp of these techniques ensures that even the most complex logarithmic puzzles can be untangled with ease.