Problem 41 - 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 \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 5+\log 2 $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\log 5 + \log 2\) is 1.

Step 1: Use the properties of logarithms

Start by applying the product property of logarithms to the expression \(\log 5 + \log 2\). This will allow you to write the expression as a single logarithm. According to the product property, the sum of the logarithms of two numbers equals the logarithm of their product i.e., for any positive real numbers a and b, \(\log a + \log b = \log (ab)\). Applying this property to the given expression gives \(\log 5 + \log 2 = \log (5*2)\).

Step 2: Simplify the Expression

Continue by evaluating the product inside the logarithm. The result will be \(\log (5*2)=\log 10\).

Step 3: Evaluate the logarithm

The logarithm base 10 of 10 is 1, because 10 raised to the power 1 is 10. The base of the logarithm in this problem is not specified which in mathematical terms implies that it is 10. Hence, the value of \(\log 10\) simplifies to 1, hence the answer to the problem is 1.

Key Concepts

Logarithmic Expressions

Logarithmic expressions represent the power to which a number, called the base, must be raised to produce a given number. They are written in the form \(\log_b a\), where \(b\) is the base and \(a\) is the number you're taking the logarithm of. Logarithms are the inverse operations of exponentiation and play a crucial role in simplifying complex mathematical equations into more manageable forms.

Understanding logarithms requires familiarity with certain definitions and rules, such as the concept that \(\log_b b = 1\) because any number raised to the first power is itself. Also important is the notion that \(\log_b 1 = 0\) since any number raised to the power of zero equals one. Grasping these fundamental principles helps in constructing and deconstructing logarithmic expressions effectively.

Product Property of Logarithms

The product property of logarithms is a powerful tool for simplifying logarithmic expressions. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors: \(\log_b (xy) = \log_b x + \log_b y\).

This property makes it easier to deal with large numbers or expressions because it breaks them down into smaller, more manageable parts. It's particularly useful when needing to condense two or more logarithmic terms into a single term. For example, applying this property to the expression \(\log 5 + \log 2\), you would combine the logs to get \(\log (5 \times 2)\), or simply \(\log 10\). This makes calculations more straightforward and is a stepping stone towards solving more complex logarithmic equations.

Condensing Logarithms

Condensing logarithms is the process of combining multiple logarithmic terms into a single term, ultimately simplifying the expression. This technique usually involves the use of properties of logarithms, such as the product, quotient, and power properties, to join logs together.

In the example \(\log 5 + \log 2\), condensing is achieved by applying the product property discussed earlier. Often, condensing is also used to solve equations or to make expressions more suitable for graphing or further analysis. It is essential to remember that when condensing, the base of the logarithms must be the same, and only terms involving logarithms can be combined.

Evaluating Logarithms

Evaluating logarithms involves finding the numerical value of a logarithmic expression. Once an expression is condensed, as in the earlier steps, it can often be evaluated to a simpler form or an exact number.

For instance, in the context of the exercise \(\log 10\), we recognize that the base is assumed to be 10 because it is not specified. The fundamental property \(\log_b b = 1\) tells us that \(\log_{10} 10 = 1\), since 10 raised to the power of 1 is indeed 10. In evaluating logarithms, knowledge of these properties and understanding the concept that any logarithm with a base equal to the number itself simplifies to one is crucial. It is these kinds of insights that help in evaluating logarithmic expressions accurately and confidently.