Problem 1 - Intermediate Algebra

Question

Write each sum as a single logarithm. Assume that variables represent positive numbers. See Example 1 . $$ \log _{5} 2+\log _{5} 7 $$

Step-by-Step Solution

Answer

\( \log_{5} 14 \)

Step 1: Recognize the Logarithm Property Used

We need to use the property of logarithms that states: \[ \log_b (m) + \log_b (n) = \log_b (m \times n) \] This property allows us to combine the sum of two logarithms with the same base into a single logarithm.

Step 2: Identify the Values to Combine

In this problem, we are given:\( \log_{5} 2 + \log_{5} 7 \)where the base \(b\) is 5, \(m\) is 2, and \(n\) is 7. These are the terms we will combine into a single logarithm.

Step 3: Apply the Property

Using the property from Step 1, combine the two logarithms:\[ \log_{5} 2 + \log_{5} 7 = \log_{5} (2 \times 7) \] This involves multiplying 2 and 7 together.

Step 4: Simplify the Expression

Calculate the product of 2 and 7:\( 2 \times 7 = 14 \)Thus, the combined logarithm is:\[ \log_{5} 14 \]

Key Concepts

Logarithm Properties

Logarithms have several important properties that help simplify expressions. Understanding these properties makes it easier to manipulate and combine logarithms. One of the most fundamental properties is the **Product Property**. This states that the sum of two logarithms with the same base can be expressed as a single logarithm of the product of their arguments. The mathematical expression for this is:

• \( \log_b (m) + \log_b (n) = \log_b (m \times n) \)

With this property, complex logarithmic expressions are often simplified by converting a sum of logarithms into a single term. This is especially useful in calculus, algebra, and other mathematical disciplines where simplifying an expression can make solving problems much simpler.
Another crucial property is the **Quotient Property**, where the difference of two logarithms with the same base equals the logarithm of the division of the arguments. It's expressed as:

• \( \log_b (m) - \log_b (n) = \log_b \left(\frac{m}{n}\right) \)

These properties are invaluable tools for simplifying logarithmic expressions and solving logarithmic equations efficiently.

Combining Logarithms

Combining logarithms is a technique used to merge two or more logarithmic terms into a single logarithmic expression. This is particularly done when the logarithms share the same base, enabling us to utilize the properties efficiently. In the problem

• \( \log_{5} 2 + \log_{5} 7 \),

we see that both terms are logarithms of base 5.
By using the Product Property, we can combine them into one logarithmic expression:

• \( \log_{5} (2 \times 7) \).

The utility of combining logarithms lies in reducing the complexity of mathematical expressions, thus making them easier to work with. It also helps in solving logarithmic equations where having a single logarithm is more straightforward to evaluate or further manipulate compared to dealing with a sum of several logarithms.
Combining logarithms becomes particularly crucial when solving real-world problems, where measured or calculated values must be consolidated to provide a more concise representation.

Logarithm Simplification

Logarithm simplification involves reducing a logarithmic expression to its simplest form. This process often utilizes the properties of logarithms as tools to break down or combine terms. When given a problem like:

• \( \log_{5} 2 + \log_{5} 7 \),

we start by applying the Product Property to combine them. This results in:

• \( \log_{5} (2 \times 7) = \log_{5} 14 \).

The final step in simplification is carrying out any actual arithmetic involved in the multiplication. Here, multiplying 2 and 7 gives us 14. The resulting logarithmic expression \( \log_{5} 14 \) is much cleaner and easier to interpret.
Simplification not only makes complex expressions more manageable but is essential in computations involving exponentials and logarithms. It is used heavily in science and engineering fields where concise and efficient calculations are necessary. Beyond practical applications, simplification also aids in understanding and uncovering deeper relationships between mathematical expressions.