Problem 10 - College Algebra

Question

For the following exercises, condense to a single logarithm if possible. $$ \log _{3}(2)+\log _{3}(a)+\log _{3}(11)+\log _{3}(b) $$

Step-by-Step Solution

Answer

\(\log_3(22ab)\)

Step 1: Understanding Logarithm Addition

When you add logarithms with the same base, you can condense them by using the property of logarithms: \(\log_b(x) + \log_b(y) = \log_b(x \times y)\). This property will help us condense the expression \(\log_3(2) + \log_3(a) + \log_3(11) + \log_3(b)\).

Step 2: Applying the Property

We start by condensing the first two terms using the property: \(\log_3(2) + \log_3(a) = \log_3(2 \times a) = \log_3(2a)\).

Step 3: Continuing Condensation

Let's add the result from the previous step with the next logarithm term: \(\log_3(2a) + \log_3(11) = \log_3(2a \times 11) = \log_3(22a)\).

Step 4: Final Condensation Step

Finally, combine \(\log_3(22a)\) with the remaining term: \(\log_3(22a) + \log_3(b) = \log_3(22a \times b) = \log_3(22ab)\).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that simplify complex mathematical expressions. Understanding these properties allows us to manipulate logarithmic equations effectively. This becomes especially crucial when dealing with multiple logarithm terms.

• Product Property: The property used in the original exercise is the product law of logarithms, which states that the sum of logarithms with the same base can be condensed into a single logarithm. Formally, \(\log_b(x) + \log_b(y) = \log_b(x \times y)\).
• Quotient Property: Another handy property is the quotient law. It tells us that the difference between two logarithms can be written as a single logarithm: \(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\).
• Power Property: When a logarithm has an exponent, it can simplify by moving the exponent in front: \(\log_b(x^n) = n\log_b(x)\).

Each of these properties allows us to either combine or break down logarithmic terms, making complex equations easier to solve.

Condensing Logarithms

Condensing logarithms involves combining several logarithm terms into a single one by using the properties of logarithms. This can be particularly useful when solving equations or simplifying expressions. In the provided exercise, we deal with multiple logarithms of the same base (3) and apply the product property repeatedly.

• Start by identifying all logarithmic terms that share the same base.
• Apply the product property step-by-step to combine terms into a product under a single logarithm, following the sequence of operations.
• Continue this process until all terms are combined into one logarithmic expression.

In our exercise, four terms \(\log_3(2), \log_3(a), \log_3(11), \log_3(b)\) are condensed in stages, resulting in \(\log_3(22ab)\). Combining terms in stages helps to keep track of calculations and ensures accuracy.

Logarithmic Identities

Logarithmic identities are crucial tools in both simplifying expressions and solving equations involving logarithms. They provide a way to exchange logarithmic expressions for equivalent forms that might be easier to handle.

• Change of Base Formula: This allows for rewriting a logarithm in terms of logs of another base: \(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\). This is particularly useful when calculators only provide logarithms of certain bases.
• One and Zero Identities: Logarithms have simple identities based on powers of 1 and 0, such as \(\log_b(1) = 0\) and \(\log_b(b) = 1\).
• Inverse Logarithms: The exponential and logarithmic functions are inverse operations, meaning \(b^{\log_b(x)} = x\) and \(\log_b(b^x) = x\).

By leveraging these identities, it's possible to transform and simplify complex logarithmic forms, making them more manageable and easier to solve.