Problem 53 - College Algebra

Question

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 2 \log _{b} x+3 \log _{b} y $$

Step-by-Step Solution

Answer

The condensed form of the given logarithmic expression is \(\log_b x^2y^3\).

Step 1: Apply the Power Rule

Implementing the power rule of logarithms: \(a\log_b x = \log_b x^a\), gives \(2 \log _{b} x+3 \log _{b} y = \log_b x^2 + \log_b y^3\)

Step 2: Apply the Product rule to Logarithms

The product rule of logarithms: \(\log_b x + \log_b y = \log_b(xy)\) can now be used to condense the expression into a single logarithm, resulting in \(\log_b x^2y^3\).

Key Concepts

Logarithmic Expressions

Logarithms are used to express the power to which a number, known as the base, must be raised to produce another number. When it comes to logarithmic expressions, it's important to recognize their connection to exponents and to understand how to work with them using various properties of logarithms. A simple logarithm looks like this: \( \log_b x \), where 'b' is the base and 'x' is the number we're interested in. The expression means that 'b' raised to what power gives us 'x'.

Dealing with logarithmic expressions often requires simplifying them. This simplification can make complex calculations more manageable and understand the relationship between different logarithmic terms. As seen in the problem from the textbook, working with multiple logarithmic terms can often lead us to condense them into a single expression, which offers a much cleaner and more efficient form for further calculations or evaluations.

In the given exercise, we're initially presented with two separate logarithmic terms: \(2 \log_b x\) and \(3 \log_b y\). The goal is to combine these two into one single logarithm with a coefficient of 1, which means we'll need to use specific rules and properties of logarithms to do so.

Power Rule of Logarithms

One of the key properties used in simplifying logarithmic expressions is the power rule of logarithms. The beauty of this rule lies in its ability to transform a coefficient outside the logarithm into an exponent on the argument inside the logarithm. The power rule can be formally expressed as follows: \(a \log_b x = \log_b x^a\).

This rule tells us that you can 'move' a multiplier on the logarithm inside as the power of its argument. For example, in our exercise, we applied the power rule to \(2 \log_b x\) and \(3 \log_b y\), transforming them into \(\log_b x^2\) and \(\log_b y^3\), respectively. Implementing this rule is crucial in simplifying expressions to a form where they can be easily combined or compared. It's essential for students to feel comfortable applying the power rule, as it's a stepping stone to mastering the manipulation of logarithms for algebraic operations.

Product Rule of Logarithms

Once logarithms have been rewritten using the power rule, we often want to combine them into a single term, which requires another property known as the product rule of logarithms. This rule is incredibly useful when you have multiple logarithmic terms that need to be condensed. It states that the sum of two logarithms with the same base is the logarithm of the product of their arguments: \(\log_b x + \log_b y = \log_b(xy)\).

By applying the product rule to the power-adjusted expressions from our example, \(\log_b x^2 + \log_b y^3\), we are able to condense them into \(\log_b (x^2y^3)\). This rule makes complex logarithmic expressions much more straightforward. When using the product rule, it's important to remember that the bases of the logarithms must be the same for it to be applied. Additionally, it emphasizes the multiplicative relationship between terms within logarithms, making the evaluation of such expressions much simpler. Understanding and applying the product rule correctly is a fundamental skill in algebra and will serve students well in various mathematical contexts.