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Chapter 4: Problem 17

Solve the equation. $$\log _{6}(2 x-3)=\log _{6} 24-\log _{6} 3$$

Short Answer

Expert verified
The solution to the equation is \(x = 5.5\).

Step by step solution

01

Apply Logarithm Quotient Rule

The right side of the equation \( \log_{6} 24 - \log_{6} 3 \) can be simplified using the logarithm quotient rule: \( \log_{b} (m) - \log_{b} (n) = \log_{b} \left( \frac{m}{n} \right) \). Thus, \( \log_{6} 24 - \log_{6} 3 = \log_{6} \left( \frac{24}{3} \right) = \log_{6} 8 \). The equation now becomes \( \log_{6} (2x - 3) = \log_{6} 8 \).
02

Equate the Arguments

Since the bases of the logarithms on both sides are equal (base 6) and the logs are equal, the arguments must be equal. Therefore, we have \(2x - 3 = 8\).
03

Solve for x

Solve the equation \(2x - 3 = 8\) by isolating \(x\). Start by adding 3 to both sides: \(2x = 11\). Then divide both sides by 2 to find \(x\): \(x = \frac{11}{2} = 5.5\).

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Key Concepts

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

Logarithm Quotient Rule
The logarithm quotient rule is an essential tool for simplifying expressions involving logarithms. It states that the log of a quotient is equal to the difference of the logs of the numerator and the denominator. This rule can be written as:\[ \log_b (\frac{m}{n}) = \log_b m - \log_b n \]In the given exercise, we applied this rule to the right-hand side of the equation: \[ \log_6 24 - \log_6 3 \]By using the logarithm quotient rule, it simplifies to:\[ \log_6 \left( \frac{24}{3} \right) = \log_6 8 \]Here are a few key steps when using the quotient rule:
  • Ensure both logs have the same base.
  • Directly substitute the quotient of the numbers into one logarithm.
  • Simplify any fractions where possible.
The rule helps in reducing complexity and making equations easy to solve.
Solving Equations
Once we've simplified the logarithmic equation using the quotient rule, the next crucial step is solving equations. Our equation becomes:\[ \log_6 (2x - 3) = \log_6 8 \]When logarithms with identical bases are equal, their arguments must also be equal. This concept allows us to bypass the logarithmic form and equate the arguments directly:\[ 2x - 3 = 8 \]To solve this equation, we can use simple algebraic steps:
  • Add 3 to both sides to isolate the term with x: \( 2x = 11 \).
  • Divide both sides by 2 to solve for x: \( x = \frac{11}{2} = 5.5 \).
Understanding these steps highlights that simplifying logarithmic equations often results in straightforward algebraic equations.
Base of Logarithms Function
The base of a logarithm is a critical component of the logarithmic function. For our purposes, the base is 6, as seen in \( \log_6 \). Here's why the base is important:
  • The base determines the number by which you raise a power to achieve a specific amount. For instance, \( \log_6 36 = 2 \) because \( 6^2 = 36 \).
  • When solving logarithmic equations, equal bases allow for the comparison of arguments directly.
Choosing the correct base is vital when working with logarithmic functions as it influences the overall solutions and can simplify process through same base comparison, as in our exercise. Once both sides of a logarithmic equation have logarithms with the same base, the equation typically reduces to a direct comparison of their arguments.

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Most popular questions from this chapter

Exer. \(43-46:\) Use natural logarithms to solve for \(x\) in terms of \(y\) $$y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$

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Bird calls decrease in intensity (loudness) as they travel through the atmosphere. The farther a bird is from an observer, the softer the sound. This decrease in intensity can be used to estimate the distance between an observer and a bird. A formula that can be used to measure this distance is $$I=I_{0}-20 \log d-k d \text { provided } 0 \leq I \leq I_{0}$$ where \(I_{0}\) represents the intensity (in decibels) of the bird at a distance of one meter (\(I_{0}\) is often known and usually depends only on the type of bird), I is the observed intensity at a distance \(d\) meters from the bird, and \(k\) is a positive constant that depends on the atmospheric conditions such as temperature and humidity. Given \(I_{0}, I,\) and \(k,\) graphically estimate the distance \(d\) between the bird and the observer. $$I_{0}=70, \quad I=20, \quad k=0.076$$

Urban population density An urban density model is a formula that relates the population density \(D\) (in thousands/mi \(^{2}\) ) to the distance \(x\) (in miles) from the center of the city. The formula \(D=a e^{-b x}\) for central density \(a\) and coefficient of decay \(b\) has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970 , \(a=5.5\) and \(b=0.10 .\) At approximately what distance was the population density 2000 per square mile?

Glottochronology is a method of dating a language at a particular stage, based on the theory that over a long period of time linguistic changes take place at a fairly constant rate. Suppose that a language originally had \(N_{0}\) basic words and that at time \(t,\) measured in millennia (1 millennium = 1000 years), the number \(N(t)\) of basic words that remain in common use is given by \(N(t)=N_{0}(0.805)^{t}\). (a) Approximate the percentage of basic words lost every 100 years. (b) If \(N_{0}=200,\) sketch the graph of \(N\) for \(0 \leq t \leq 5\).
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