Problem 24 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} 2+\log _{5} y^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{5}(2y^2) \).

Step 1: Identify the Logarithmic Properties to Use

We need to combine two logarithmic expressions into a single logarithm. Here, both terms share the same base, which is 5. For expressions with the same base, we can use the 'Product Rule for Logarithms', which states that \( \log_b(x) + \log_b(y) = \log_b(xy) \).

Step 2: Rewrite the Expression Using the Product Rule

Applying the product rule for logarithms, we can combine the two logarithms as follows: \( \log_{5}(2) + \log_{5}(y^2) = \log_{5}(2 \, y^2) \). This is because multiplying the bases of the logarithms (2 and \(y^2\)) allows us to combine them into a single logarithmic expression.

Step 3: Simplify the Expression

Simplify the expression inside the logarithm. Here, \(2 \cdot y^2\) is already in its simplest form inside the product rule, so the final expression becomes \( \log_{5}(2y^2) \). This is now our single logarithmic expression.

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms to express mathematical relationships. A logarithm answers the question: "To what power must a base be raised to produce a given number?" Think of it as the reverse of an exponentiation. For example, if we have the expression \( \log_{b}(x) \), it means you're asking what power you need to raise \( b \) to in order to get \( x \). When dealing with logarithmic expressions, it's important to pay attention to the base of the logarithm. The expression \( \log_{5}(2) \) tells us we're using 5 as the base. This is consistent throughout the logarithmic expressions in the problem at hand.Logarithmic expressions can often be combined or expanded using various logarithmic rules that make calculations easier or help reduce complex expressions into simpler forms. These rules are essential when you want to perform operations like addition, subtraction, or even integration and differentiation with logarithms.

Product Rule for Logarithms

The product rule for logarithms is an essential identity in logarithmic operations. It states that when you add two logarithms that have the same base, you can combine them by multiplying their arguments. Mathematically, this can be stated as:\[\log_{b}(x) + \log_{b}(y) = \log_{b}(xy)\]This rule is incredibly helpful because it allows you to turn a sum of logarithms into a single, simpler logarithmic expression. For example, in the given exercise, we have two logarithms: \( \log_{5}(2) \) and \( \log_{5}(y^2) \). Since they both have the base 5, we can use the product rule to combine them into a single logarithmic expression:\[\log_{5}(2) + \log_{5}(y^2) = \log_{5}(2 y^2)\]The product rule simplifies calculations and helps when you need to evaluate or solve logarithmic equations because it reduces the number of terms you have to work with.

Combining Logarithms

Combining logarithms is a useful technique when simplifying expressions or solving equations. It involves using logarithmic rules like the product, quotient, and power rules to merge several logarithms into one. This makes expressions simpler and easier to manipulate.When combining logarithms using the product rule, as we've seen, you multiply the inner terms of each logarithmic expression. In the example \( \log_{5}(2) + \log_{5}(y^2) \), these combined as:- Multiply the numbers or expressions within the logarithms: \( 2 \times y^2 \)- Result in a single expression: \( \log_{5}(2y^2) \)Other useful rules include:

• The **quotient rule**, which allows you to combine logarithms when you have subtraction: \( \log_{b}(x) - \log_{b}(y) = \log_{b}(\frac{x}{y}) \).
• The **power rule**, which simplifies logarithmic expressions involving exponents: \( \log_{b}(x^a) = a \cdot \log_{b}(x) \).

Understanding how to effectively combine and manipulate logarithms is crucial in higher algebra and calculus, where solving for unknowns often involves logarithmic functions. By mastering these techniques, you can tackle a wide range of mathematical problems with confidence.