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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

Newton's Law of Cooling. Suppose that a body with temperature \(T_{1}\) is placed in surroundings with temperature \(T_{0}\) different from that of \(T_{1}\). The body will either cool or warm to temperature \(T(t)\) after time \(t,\) in minutes, where $$T(t)=T_{0}+\left(T_{1}-T_{0}\right) e^{-i t}$$,A chilled gelatin salad that has a temperature of \(43^{\circ} \mathrm{F}\) is taken from the refrigerator and placed on the dining room table in a room that is \(68^{\circ} \mathrm{F}\). After \(12 \mathrm{min}\), the temperature of the salad is \(55^{\circ} \mathrm{F}\). What will the temperature of the salad be after 20 min?

Given that \(\log _{b} 2=0.693, \log _{b} 3=1.099,\) and \(\log _{b} 5=1.609,\) find each of the following, if possible. Round the answer to the nearest thousandth. $$\log _{b} 125$$

Newton's Law of Cooling. Suppose that a body with temperature \(T_{1}\) is placed in surroundings with temperature \(T_{0}\) different from that of \(T_{1}\). The body will either cool or warm to temperature \(T(t)\) after time \(t,\) in minutes, where $$T(t)=T_{0}+\left(T_{1}-T_{0}\right) e^{-i t}$$,A cup of coffee with temperature \(105^{\circ} \mathrm{F}\) is placed in a freezer with temperature \(0^{\circ} \mathrm{F}\). After \(5 \mathrm{min}\), the temperature of the coffee is \(70^{\circ} \mathrm{F}\). What will its temperature be after 10 min?

Graph the function by substituting and plotting points. Then check your work using a graphing calculator. $$f(x)=3^{-x}$$

Express as a single logarithm and, if possible, simplify. $$\log _{a}\left(a^{10}-b^{10}\right)-\log _{a}(a+b)$$
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