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Chapter 5: Problem 44

Solve for \(x\) in terms of \(b: \log _{b}(1-3 x)=3+\log _{b} x.\)

Short Answer

Expert verified
The solution is: \( x = \frac{1}{b^3 + 3} \).

Step by step solution

01

Simplify the equation using the properties of logarithms

First, we'll take advantage of the properties of logarithms, specifically the product rule and power rule. The original equation is: \[\log_b(1-3x) = 3 + \log_b(x)\]This can be rewritten as:\[\log_b(1-3x) = \log_b(b^3) + \log_b(x)\]
02

Apply the product rule of logarithms

Utilize the product rule of logarithms which states that \( \log_b(a) + \log_b(c) = \log_b(ac) \). So we have:\[\log_b(b^3) + \log_b(x) = \log_b(b^3 \cdot x)\]Now equate the logarithms:\[\log_b(1-3x) = \log_b(b^3 \cdot x)\]
03

Set the arguments of the logarithms equal

Since the logarithms of the same base are equal, their arguments must also be equal:\[1-3x = b^3 \cdot x\]
04

Solve for \( x \)

Rearrange the equation to solve for \( x \):\[1 = b^3 \cdot x + 3x\]\[1 = x(b^3 + 3)\]\[x = \frac{1}{b^3 + 3}\]

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Key Concepts

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

Properties of Logarithms
Logarithms have unique properties that make them incredibly useful in mathematics. Understanding these properties is key when solving equations involving logarithms. Here are some fundamental properties:

  • Product Rule: This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors: \( \log_b(a \cdot c) = \log_b(a) + \log_b(c) \).
  • Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \log_b\left( \frac{a}{c} \right) = \log_b(a) - \log_b(c) \).
  • Power Rule: The logarithm of a power is the exponent times the logarithm of the base: \( \log_b(a^c) = c \cdot \log_b(a) \).

In the given exercise, these properties allow us to manipulate the logarithmic equation into a more solvable form. Utilizing them is often necessary not just for transforming the equation but also for providing insights into the relationships between the quantities involved.
Product Rule of Logarithms
The product rule of logarithms is one of the most versatile and handy tools in working with logarithmic expressions. According to the product rule, \( \log_b(a) + \log_b(c) = \log_b(a \cdot c) \). This rule allows us to combine two separate logarithmic expressions into a single one.

This is especially useful when the equation involves the addition of logarithms, as in this exercise. By taking the expression \( \log_b(x) + 3 \), we first express \( 3 \) as \( \log_b(b^3) \), transforming it into a compatible form with \( \log_b(x) \). The sum becomes \( \log_b(b^3 \cdot x) \), streamlining the equation and setting us up for future simplification steps.

Remember that the product rule requires both terms to have the same base, which allows us to apply it confidently.
Logarithmic Equations
Logarithmic equations are equations that involve the logarithm of a variable expression. Solving these equations often involves using properties of logarithms to combine, simplify, or otherwise transform the equation into a more solvable form.

In the given exercise, the equation \( \log_b(1-3x) = 3 + \log_b(x) \) involves logarithms of the variable \( x \). To solve for \( x \), we must navigate the intricacies of logarithmic manipulation.

Typically, we aim to isolate the logarithmic expression and eliminate logarithms by converting them into exponential form. Once simplified, it becomes more straightforward to solve the equation using regular algebraic methods. This process may include using properties such as combining logarithms into a single expression or equating the arguments of logarithms when bases agree.
Algebraic Manipulation
Algebraic manipulation is essential for solving equations and typically involves rearranging terms, factoring expressions, or using identities to simplify or solve equations. When dealing with logarithmic equations, algebraic manipulation allows us to isolate the variable after the logarithm has been simplified.

In this problem, after consolidating the logarithmic terms, the argument equality \( 1 - 3x = b^3 \cdot x \) is reached. This step requires rearranging terms to further isolate \( x \). By bringing all terms involving \( x \) to one side and constant terms to the other, we eventually express \( x \) in terms of constants and known values.

Finally, by dividing through by the coefficient of \( x \), we isolate \( x \) fully, resulting in \( x = \frac{1}{b^3 + 3} \).

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Most popular questions from this chapter

Exercises \(55-60\) introduce a model for population growth that takes into account limitations on food and the environment. This is the logistic growth model, named and studied by the nineteenth century Belgian mathematician and sociologist Pierre Verhulst. (The word "logistic" has Latin and Greek origins meaning "calculation" and "skilled in calculation," respectively. However, that is not why Verhulst named the curve as he did. See Exercise 56 for more about this.) In the logistic model that we "I study, the initial population growth resembles exponential growth. But then, at some point owing perhaps to food or space limitations, the growth slows down and eventually levels off, and the population approaches an equilibrium level. The basic equation that we'll use for logis- tic growth is where \(\mathcal{N}\) is the population at time \(t, P\) is the equilibrium population (or the upper limit for population), and a and b are positive constants. $$\mathcal{N}=\frac{P}{1+a e^{-b t}}$$ The following figure shows the graph of the logistic function \(\mathcal{N}(t)=4 /\left(1+8 e^{-t}\right) .\) Note that in this equation the equilibrium population \(P\) is 4 and that this corresponds to the asymptote \(\mathcal{N}=4\) in the graph. (a) Use the graph and your calculator to complete the following table. For the values that you read from the graph, estimate to the nearest \(0.25 .\) For the calculator values, round to three decimal places. (b) As indicated in the graph, the line \(\mathcal{N}=4\) appears to be an asymptote for the curve. Confirm this empirically by computing \(\mathcal{N}(10), \mathcal{N}(15),\) and \(\mathcal{N}(20) .\) Round each answer to eight decimal places. (c) Use the graph to estimate, to the nearest integer, the value of \(t\) for which \(\mathcal{N}(t)=3\) (d) Find the exact value of \(t\) for which \(\mathcal{N}(t)=3 .\) Evaluate the answer using a calculator, and check that it is consistent with the result in part (c). TABLE AND GRAPH CANT COPY

[From Example 9(b)] Solve the inequality \(x^{2}-2 x-24 \leq 0 .\) You should find that the solution set is the closed interval [-4,6].

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