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Chapter 8: Problem 17

Write each logarithmic expression as a single logarithm. \(\log _{6} 5+\log _{6} x\)

Short Answer

Expert verified
The single logarithm for the given expressions is \(\log_6 (5 * x)\).

Step by step solution

01

Identify the Given Logarithms

The given logarithmic expressions are \(\log _{6} 5\) and \(\log _{6} x\). Notice that these logarithms have the same base, which is 6.
02

Apply the Logarithmic Property

Applying the log property \(\log_a b + \log_a c = \log_a (b*c)\), we can combine the expressions as follows: \(\log _{6} 5+\log _{6} x = \log_6 (5 * x)\)
03

Simplify the expression

To get our final answer, simplify the multiplication inside the logarithm as much as possible. In this case, it remains as is. Thus, the combined single logarithmic expression is \(\log_6 (5 * x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Logarithms
Logarithms have specific properties that make them extremely useful in math, particularly for simplifying complex problems. One important property is the **product rule** of logarithms, which helps us when we want to combine log expressions.

The product rule states that if you have two logs with the same base, you can combine them into a single log. Mathematically, it is represented as:

\[\log_a b + \log_a c = \log_a (b \times c)\]

Here are some key points to remember about the properties of logarithms:
  • **Base Consistency:** When using these properties, the logarithms must always have the same base.
  • **Combination Strength:** The product rule is handy when adding two logs. You turn them into a multiplication expression inside one log.
  • **Rule Flexibility:** This rule applies to any numbers as long as the bases match.
These properties simplify calculations, especially in algebra and calculus, making it easier to deal with exponentials and complex equations.
Combining Logarithms
Combining logarithms is about using properties like the product rule to merge several log expressions into one. This is valuable because it reduces the number of steps needed to solve problems, thereby making equations simpler and more straightforward.

To combine logs, first ensure they have the same base. In this exercise, both logs have a base of 6, which means they can be combined. Using the product rule, we have:

\[\log_6 5 + \log_6 x = \log_6 (5 \times x)\]

List of steps to combine logs:
  • Check that the logs have the same base.
  • Use the appropriate property, like the product rule, to merge them.
  • Simplify the resulting expression if possible.
Combining logs simplifies complex expressions and can be applied in various mathematical tasks, making it an essential skill.
Single Logarithm Expression
Writing a logarithmic expression as a single log is the final goal when simplifying expressions using the properties of logarithms. A single log expression is not only simpler to read but also easier to work with, particularly in equations and more advanced math problems.
After applying the product rule in our context, we get:
\[\log_6 (5 \times x)\]

Here's why having a single log expression is beneficial:
  • **Simplification:** Reduces the complexity of expressions, thus making mathematical operations simpler.
  • **Efficiency:** Solving equations becomes faster since there are fewer expressions to deal with.
  • **Clarity:** A single expression is often clearer, which reduces errors in calculation.
In summary, mastering how to write complex logarithmic expressions as a single log provides clarity and efficiency in solving mathematical tasks.

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Most popular questions from this chapter

Use the Change of Base Formula to evaluate each expression. Then convert it to a logarithm in base \(8 .\) $$ \log _{3} 33 $$

Mental Math Solve each equation. $$ 25^{x}=\frac{1}{5} $$

Solve each equation. $$ \log 9-\log x+1=6 $$

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Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \log _{7} 3 x=3 $$
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