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Chapter 5: Problem 91

Express as a sum or a difference of logarithms. $$\log _{a} \frac{x-y}{\sqrt{x^{2}-y^{2}}}$$

Short Answer

Expert verified
\( \log_{a} (x-y) - \frac{1}{2} \log_{a} (x^{2}-y^{2}) \)

Step by step solution

01

- Apply the Quotient Rule

Use the quotient rule of logarithms: \( \log_{a} \left( \frac{x}{y} \right) = \log_{a} x - \log_{a} y \). Here, we can write it as \( \log_{a} \frac{x - y}{\sqrt{x^{2} - y^{2}}} = \log_{a} (x-y) - \log_{a} (\sqrt{x^{2} - y^{2}}) \)
02

- Apply the Power Rule

Use the power rule of logarithms: \( \log_{a} (x^{n}) = n \log_{a} x \). Note that \( \sqrt{x^{2}-y^{2}} = (x^{2}-y^{2})^{\frac{1}{2}} \). So, \( \log_{a} (\sqrt{x^{2} - y^{2}}) = \log_{a} ((x^{2} - y^{2})^{\frac{1}{2}}) = \frac{1}{2} \log_{a} (x^{2} - y^{2}) \)
03

- Combine the Results

Combine the results from the previous steps: \( \log_{a} (x-y) - \log_{a} (\sqrt{x^{2} - y^{2}}) = \log_{a} (x-y) - \frac{1}{2} \log_{a} (x^{2} - y^{2}) \)

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Key Concepts

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

Quotient Rule
The quotient rule in logarithms is a helpful property. It allows us to simplify the logarithm of a fraction. The rule states: \(\text{log}_a \frac{m}{n} = \text{log}_a m - \text{log}_a n\). This means if you have the logarithm of a division, you can split it into a difference of two logarithms.
Let's apply this rule to our problem. We start with: \(\text{log}_a \frac{x-y}{\text{sqrt}(x^2 - y^2)}\). Using the quotient rule, we can rewrite it as: \(\text{log}_a (x-y) - \text{log}_a (\text{sqrt}(x^2 - y^2))\).
This step simplifies our expression and sets us up to further break it down.
Power Rule
The power rule in logarithms helps manage exponents. It states: \(\text{log}_a (m^n) = n \text{log}_a m\). This means that the exponent can be brought in front of the logarithm as a multiplier.
In the given problem, after applying the quotient rule, we had: \(\text{log}_a (\text{sqrt}(x^2 - y^2))\). Since \(\text{sqrt}(x^2 - y^2)\) is the same as \((x^2 - y^2)^{\frac{1}{2}}\), we can use the power rule. This transforms:
\(\text{log}_a (\text{sqrt}(x^2 - y^2)) = \text{log}_a ((x^2 - y^2)^{\frac{1}{2}}) = \frac{1}{2} \text{log}_a (x^2 - y^2)\).
Applying the power rule simplifies the expression by moving the fraction exponent in front.
Logarithmic Expressions
Logarithmic expressions often combine multiple logarithmic properties. This helps in breaking down complex logarithm functions into simpler parts.
In our exercised problem, after applying the quotient and power rules, we have: \(\text{log}_a (x-y) - \frac{1}{2} \text{log}_a (x^2 - y^2)\).
This result shows how logarithm properties simplify complex expressions into sums or differences of simpler logarithms.
Remember these core rules:
  • The quotient rule allows subtraction of logarithms.
  • The power rule allows exponent handling by converting them to multipliers.

Understanding these concepts helps in dealing with various logarithmic expressions encountered in math.

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Most popular questions from this chapter

In Exercises \(77-80\) : a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or a minimum value and find that value.[ 3.3] $$G(x)=-2 x^{2}-4 x-7$$

Solve using any method. $$\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=3$$

Use a graphing calculator to find the point \((s)\) of intersection of the graphs of each of the following pairs of equations. $$y=\left|1-3^{x}\right|, y=4+3^{-x^{2}}$$

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Suppose that \(\log _{a} x=2 .\) Find each of the following. Simplify: $$\log _{10} 11 \cdot \log _{11} 12 \cdot \log _{12} 13 \cdots \log _{998} 999 \cdot \log _{999} 1000$$
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