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Chapter 9: Problem 70

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write as a single logarithm: \(\log _{7} x+3 \log _{7} y-\log _{7}(x+y)\)

Short Answer

Expert verified
\( \log_{7}(\frac{x \cdot y^3}{x + y}) \)

Step by step solution

01

- Use the power rule for logarithms

Apply the power rule of logarithms: The power rule states that 3 \log_{7}(y) = \log_{7}(y^{3}). So rewrite the given expression: \( \log_{7}(x) + 3 \log_{7}(y) - \log_{7}(x + y) = \log_{7}(x) + \log_{7}(y^3) - \log_{7}(x + y) \)
02

- Use the product rule for logarithms

Apply the product rule of logarithms: The product rule states that \(\log_{b}(a) + \log_{b}(c) = \log_{b}(a \cdot c)\). So combine \log_{7}(x) and \log_{7}(y^3): \( \log_{7}(x) + \log_{7}(y^3) = \log_{7}(x \cdot y^3) \)
03

- Use the quotient rule for logarithms

Apply the quotient rule of logarithms: The quotient rule states that \(\log_{b}(a) - \log_{b}(c) = \log_{b}(\frac{a}{c})\). So combine \log_{7}(x \cdot y^3) and \log_{7}(x + y): \( \log_{7}(x \cdot y^3) - \log_{7}(x + y) = \log_{7}(\frac{x \cdot y^3}{x + y}) \)

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Key Concepts

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

Power Rule of Logarithms
The power rule of logarithms is a fundamental rule that helps make complex logarithmic expressions simpler. When you have a logarithm with an exponent, you can move the exponent in front of the logarithm. This rule can be expressed as: \ \( \log_{b}(a^{c}) = c \log_{b}(a)\ \).
In our exercise, we used this rule to rewrite the term \ \( 3 \log_{7}(y) \). Applying the power rule, we get: \ \( 3 \log_{7}(y) = \log_{7}(y^{3}) \).
This transformation simplifies handling logarithmic expressions by reducing the number of terms in the equation.
Product Rule of Logarithms
The product rule of logarithms is useful when you need to combine the logarithms of multiple numbers. According to this rule: \ \( \log_{b}(a) + \log_{b}(c) = \log_{b}(a \cdot c)\ \).
This allows you to merge two separate logarithms into a single logarithm of the product of their arguments.
In the step-by-step solution, we applied this rule to \ \( \log_{7}(x) + \log_{7}(y^{3}) = \log_{7}(x \cdoty^{3}) \).
By using this rule, we combined the logarithms into a single term, making our expression more manageable.
Quotient Rule of Logarithms
The quotient rule of logarithms is handy for combining logarithms that involve division. This rule states: \ \( \log_{b}(a) - \log_{b}(c) = \log_{b}( \frac{a}{c} )\ \). Combining logarithms this way can simplify expressions and make them more straightforward to work with.
In our exercise, we used the quotient rule on \ \( \log_{7}(x \cdot y^{3}) - \log_{7}(x + y)\ \). Applying the quotient rule, we get: \ \( \log_{7}( \frac{x \cdot y^{3}}{x + y} )\ \).
This final step combined our terms into a single logarithmic expression, which is much easier to interpret and use for further calculations.

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Most popular questions from this chapter

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=4 ; \quad T=\frac{\pi}{2} \text { seconds } $$

In Problems 17-32, solve each triangle. $$ a=3, \quad b=4, \quad C=40^{\circ} $$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=-9 \sin \left(\frac{1}{4} t\right) $$

In Problems \(47-52,\) the function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-20 e^{-0.7 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^{2}-\frac{0.49}{1600}} t\right) $$

How would you explain simple harmonic motion to a friend? How would you explain damped motion?
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