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Chapter 9: Problem 31

Express as an equivalent expression that is a single logarithm. $$\log _{a} x-\log _{a} y$$

Short Answer

Expert verified
\[ \log_a\left(\frac{x}{y}\right) \]

Step by step solution

01

- Apply the Logarithm Subtraction Rule

The subtraction of two logarithms with the same base can be written as a single logarithm using the rule: \[ \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \]
02

- Substitute the Values

In this case, we have \log_a(x) - \log_a(y). Using the logarithm subtraction rule from Step 1, we can write: \[ \log_a\left(\frac{x}{y}\right) \]

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Key Concepts

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

Logarithm Subtraction Rule
Understanding the logarithm subtraction rule is crucial for simplifying expressions. This rule states that when you subtract one logarithm from another, if both logarithms have the same base, you can combine them into a single logarithm.

The formula looks like this: \[ \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \] This means if you have \log_b(M) and \log_b(N), and you subtract these logarithms, you can represent them as a quotient inside a single logarithm with the base \( b \).

For example:
  • If we have \log_2(8) - \log_2(4), this simplifies to \log_2(\frac{8}{4}) which is \log_2(2).
  • If we have \log_3(81) - \log_3(9), this simplifies to \log_3(\frac{81}{9}) which is \log_3(9).
This rule helps combine logarithmic expressions into a more compact and simpler form.
Single Logarithm
A single logarithm is an expression that combines multiple logarithms into one, using properties and rules of logarithms. Converting multiple logarithms into one can make solving equations easier and the expressions cleaner.

In the given exercise, we start with the expression \log_a(x) - \log_a(y). By applying the logarithm subtraction rule,
we combine these into a single logarithm: \log_a(\frac{x}{y}).

Let's break it down into steps:
  • Recognize that both logarithms \log_a(x) and \log_a(y) have the same base \( a \).
  • Apply the subtraction rule: \[ \log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right) \]
Now, instead of dealing with two logarithms, we have one single logarithm: \log_a(\frac{x}{y})

This form is often more useful and simpler for further calculations or solving equations.
Base of Logarithm
The base of a logarithm is the number that a given number is raised to, to get another number. In logarithmic expressions, the base is the little number written right beside the log symbol.

For the logarithms to be combined using the properties we've discussed, they must have the same base.
For example:
  • In \log_2(8), the base is 2.
  • In \log_10(100), the base is 10 (often just written log(100), assuming a common base of 10).
In our exercise, \log_a(x) and \log_a(y) both have the base \( a \). Because of this, we can apply the subtraction rule correctly. If the bases were different, we could not combine the logarithms directly.

Understanding the base is essential when working with logarithms, as it determines how these logarithmic rules and properties can be applied.

In summary:
  • The base must be the same for combining logarithms.
  • The base is the number to which another number is raised to reach your given number.
  • Logs with different bases cannot be combined directly.

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Most popular questions from this chapter

Simplify. $$ \frac{\sqrt[3]{24 x y^{8}}}{\sqrt[3]{3 x y}}[7.4] $$

Simplify. $$ \log _{10}\left(\log _{4}\left(\log _{3} 81\right)\right) $$

Let \(f(x)=x^{3}-5 .\) Use composition of inverse functions to show that $$f^{-1}(x)=\sqrt[3]{x+5}$$

An organization determines that the cost per person \(C(x),\) in dollars, of chartering a bus with \(x\) passengers is given by $$C(x)=\frac{100+5 x}{x}$$ Determine \(C^{-1}(x)\) and explain how this inverse function could be used.

Let \(f(x)=(1-x) / x .\) Use composition of inverse functions to show that $$f^{-1}(x)=\frac{1}{x+1}$$
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