Problem 11 - Precalculus Functions and Graphs

Question

Rewrite each expression as a single logarithm. $$\log (5)+\log (3)$$

Step-by-Step Solution

Answer

\log (15) \

Step 1: Identify the Logarithm Rule

Determine which logarithm rule can be applied to the given expression \( \log (5) + \log (3) \). The rule to combine two logarithms into a single logarithm is the product rule, which states that \( \log_a (x) + \log_a (y) = \log_a (xy) \).

Step 2: Apply the Product Rule

Use the product rule to combine \( \log (5) \) and \( \log (3) \). According to the product rule, \( \log (5) + \log (3) = \log (5 \times 3) \).

Step 3: Simplify the Expression

Simplify the expression inside the logarithm to complete the step of rewriting the expression as a single logarithm. \( 5 \times 3 = 15 \), so the expression becomes \log (15)\.

Key Concepts

product rule

The product rule is a key concept in logarithms that helps simplify expressions by combining two logarithmic terms. According to the product rule for logarithms, when you have the sum of two logarithms with the same base, you can combine them into a single logarithm by multiplying their arguments. Mathematically, it is expressed as \( \log_a (x) + \log_a (y) = \log_a (xy) \).
This rule is very handy when you need to simplify many logarithmic expressions.
For example, to simplify \( \log(5) + \log(3) \), we apply the product rule:
\( \log(5) + \log(3) = \log(5 \times 3) = \log(15) \).
This way, you have turned two separate logarithms into a single one. Simple, right?

combining logarithms

Combining logarithms involves using rules like the product rule to turn multiple logarithmic terms into a single term. This process makes equations easier to handle and solve.
When you are given a problem with multiple logarithms that have the same base, look for opportunities to combine them. The product rule is a primary tool you can use in these scenarios.
For instance, combining \( \log(5) \) and \( \log(3) \) means employing the product rule mentioned before:
\( \log(5) + \log(3) \) becomes \( \log(15) \).
More advanced cases may require you to combine logarithms in multiple steps or using other rules like the quotient rule or power rule. But for basic cases, the product rule is often sufficient.

simplifying logarithmic expressions

Simplifying logarithmic expressions is all about reducing complexity by combining multiple logarithms into a single term. This is crucial for solving equations more easily and understanding the properties of logarithms.
Take the previous example: starting with \( \log(5) + \log(3) \), applying the product rule yields \( \log(15) \).
Here’s a quick checklist for simplifying:

• Identify the logarithm rules that are applicable: product rule for sums, quotient rule for differences, and power rule if exponents are involved.
• Combine logarithms step-by-step, simplifying as you go.
• Always ensure the base of the logarithms is the same before combining them.

By following this systematic approach, you can tackle more complex problems with confidence.