Problem 88 - College Algebra and Calculus: An Applied Approach

Question

Condense the expression to the logarithm of a single quantity.\(\log _{5} y+\log _{5} x\)

Step-by-Step Solution

Answer

\(\log _{5}(xy)\)

Step 1: Identify the rule

First, identify the rule of logarithms that states the sum of the logarithms equals the logarithm of their product. Written mathematically, this is: \(\log_b(m) + \log_b(n) = \log_b(mn)\). This rule will be applied to this exercise.

Step 2: Apply the rule

Apply this rule to the given expression \(\log _{5} y + \log _{5} x\), by considering \(m = x\) and \(n = y\). As per the rule, this yields: \(\log _{5}(xy)\).

Step 3: Write the final answer

The simplified form of the given expression is \(\log _{5}(xy)\).

Key Concepts

Logarithmic Identities

Logarithmic identities are the mathematical rules that allow you to manipulate and simplify expressions involving logarithms. They are crucial for solving equations and condensing or expanding logarithmic expressions. The fundamental logarithmic identities include:

• Product Identity: \( \log_b(m) + \log_b(n) = \log_b(mn) \)
• Quotient Identity: \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)
• Power Identity: \( \log_b(m^n) = n \cdot \log_b(m) \)

These identities are based on the properties of exponents since a logarithm is essentially an inverse operation to exponentiation. Understanding these identities helps simplify complex logarithmic expressions and solve logarithmic equations by transforming them into a more manageable form. Using these identities accurately can often turn a challenging expression into a simple calculation.

Condensing Logarithms

Condensing logarithms means combining multiple logarithmic terms into a single logarithm. This process can simplify problem-solving and makes equations easier to handle. When you have a sum or difference of logarithms, you can use logarithmic identities to combine them.
For instance, if you have terms like \( \log_b(x) + \log_b(y) \), you can use the product identity to condense it into \( \log_b(xy) \). Similarly, for the difference \( \log_b(x) - \log_b(y) \), you use the quotient identity to result in \( \log_b\left(\frac{x}{y}\right) \).
Condensing logarithms is particularly useful in solving logarithmic equations, where reducing the number of terms can significantly simplify the solving process. It can also aid in understanding and deriving exponential relationships in complex problems.

Product Rule of Logarithms

The product rule of logarithms is a key concept that states that the logarithm of a product is equal to the sum of the logarithms of its factors. This rule is formally expressed as \( \log_b(mn) = \log_b(m) + \log_b(n) \).
To apply the product rule, identify parts of an expression that can be seen as a product. For example, for the expression \( \log_b(x) + \log_b(y) \), you can condense it using the product rule to transform it into \( \log_b(xy) \). This condensing helps streamline equations by transforming multiple logarithmic terms into one.
Understanding the product rule of logarithms can greatly simplify algebraic manipulations involving logarithms, particularly in calculus and higher-level mathematics where you often encounter complex logarithmic expressions. Mastering this basic logarithmic property is essential for tackling more advanced mathematical problems.