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Chapter 3: Problem 28

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}\left(\frac{x y}{z}\right)$$

Short Answer

Expert verified
\( \log_{b}(x) + \log_{b}(y) - \log_{b}(z) \)

Step by step solution

01

Apply Logarithmic Division Rule

The logarithmic division rule states that \( \log_b\left(\frac{A}{B}\right) = \log_b A - \log_b B \). We apply this to the given expression \( \log_{b}\left(\frac{xy}{z}\right) \) such that \( A = xy \) and \( B = z \). Thus, we have: \[ \log_b\left(\frac{xy}{z}\right) = \log_b (xy) - \log_b z \].
02

Apply Logarithmic Multiplication Rule

Next, we apply the multiplication rule for logarithms, \( \log_b(AB) = \log_b A + \log_b B \). Therefore, substitute \( A = x \) and \( B = y \) into \( \log_b (xy) \), resulting in: \[ \log_b (xy) = \log_b x + \log_b y \].
03

Combine Results

Combine the results from Step 1 and 2, substituting \( \log_b (xy) = \log_b x + \log_b y \) back into the expression from Step 1. So we have: \[ \log_b\left(\frac{xy}{z}\right) = (\log_b x + \log_b y) - \log_b z \].
04

Simplify the Expression

Finally, simplify the expression by distributing the subtraction to each term, giving us: \[ \log_b x + \log_b y - \log_b z \].

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Key Concepts

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

Logarithmic Division Rule
The logarithmic division rule is a valuable tool when working with quotients in logarithmic expressions. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, it can be represented as: \[ \log_b\left(\frac{A}{B}\right) = \log_b A - \log_b B \] This rule can simplify complex logarithmic expressions by breaking them down into simpler parts. Let's apply this to the provided exercise:
  • For the expression \( \log_b\left(\frac{xy}{z}\right) \), identify \( A \) as \( xy \), and \( B \) as \( z \).
  • Then substitute those into the rule to get \( \log_b (xy) - \log_b z \).
Here, we are separating the fraction into a straightforward subtraction of two logarithms, simplifying our calculation and understanding of the expression.
Logarithmic Multiplication Rule
Once the division rule is applied, we often need to further break down logarithmic expressions using the logarithmic multiplication rule. This rule simplifies the logarithm of a product into the sum of two logarithms: \[ \log_b(AB) = \log_b A + \log_b B \] The purpose of this rule is to further simplify expressions involving products. We apply it to our intermediate result:
  • Consider \( \log_b(xy) \) from the previous section, where \( A = x \) and \( B = y \).
  • Substitute these into the multiplication rule to obtain \( \log_b x + \log_b y \).
In doing so, we express the original product \( xy \) as a sum of logarithms. This transformation is crucial for simplification and makes handling the expression easier.
Simplifying Logarithms
After applying the division and multiplication rules, the final task is to combine and simplify the logarithmic expression. Combining the results of these rules allows us to write the expression clearly and concisely. In our example, after applying the multiplication rule, we substitute back into the expression dealt with the division rule:
  • Combine \( \log_b (xy) = \log_b x + \log_b y \) with \( \log_b\left(\frac{xy}{z}\right) = \log_b (xy) - \log_b z \).
  • This results in the expression \( \log_b x + \log_b y - \log_b z \).
These steps remove unnecessary complexity and express the original logarithmic expression as a straightforward combination of logarithmic values of each component. Simplifying logarithms in this manner makes it easier to grasp the relationships within the expression, aiding deeper understanding and quicker computations.

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Most popular questions from this chapter

Money invested in an account that compounds interest continuously at a rate of \(3 \%\) a year is modeled by \(A=A_{0} e^{0.03 t},\) where \(A\) is the amount in the investment after \(t\) years and \(A_{0}\) is the initial investment. How long will it take the initial investment to double?

Explain the mistake that is made. The city of Orlando, Florida, has a population that is growing at \(7 \%\) a year, compounding continuously. If there were 1.1 million people in greater Orlando in \(2006,\) approximately how many people will there be in \(2016 ?\) Apply the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of people. Solution: Use the population growth model. \(N=N_{0} e^{r t}\). Let \(N_{0}=1.1, r=7,\) and \(t=10 . \quad N=1.1 e^{(7)(10)}\). Approximate with a calculator. \(2.8 \times 10^{30}\). This is incorrect. What mistake was made?

Refer to the following: In calculus, to find the derivative of a function of the form \(y=k^{x}\) where \(k\) is a constant, we apply logarithmic differentiation. The first step in this process consists of writing \(y=k^{x}\) in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. $$y=4^{x} \cdot 3^{x+1}$$

Explain the mistake that is made. State the domain of the logarithmic function \(f(x)=\ln |x|\) in interval notation. Solution: since the absolute value eliminates all negative numbers, the domain is the set of all real numbers. Interval notation: \((-\infty, \infty)\) This is incorrect. What went wrong?

Determine whether each statement is true or false. $$e^{\log x}=x$$
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