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Chapter 2: Problem 22

Solve the inequalities (a) \(3^{2 x+1} \leq 7\) (b) \(0.8^{x}<0.04\)

Short Answer

Expert verified
For (a), \[x \leq \frac{\log (7) - \log (3)}{2 \log(3)}\]. For (b), x < \frac{\log(0.04)}{\log(0.8)}.

Step by step solution

01

Solve the first inequality

Consider the inequality: \[3^{2x+1} \leq 7\] Take the natural logarithm on both sides to utilize the logarithmic property: \[\log(3^{2x+1}) \leq \log(7)\] Using the power rule of logarithms, bring the exponent down: \[(2x+1)\log(3)\leq \log(7)\] Solve for x by isolating it: \[2x\log(3) + \log(3)\leq \log(7)\] \[2x\log(3) \leq \log(7) - \log(3)\] Dividing both sides by the coefficient of x: \[x\leq \frac{\log(7) - \log (3)}{2\log(3)}\]
02

Solve the second inequality

Consider the inequality: \[0.8^{x}<0.04\] Take the natural logarithm on both sides to utilize the logarithmic property: \[\log(0.8^{x}) \< \log (0.04)\] Using the power rule of logarithms, bring the exponent down: \[x\log(0.8)\< \log(0.04)\] Solve for x by isolating it: \[x < \frac{\log(0.04)}{\log(0.8)} \]

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Key Concepts

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

Inequality Solving
Inequalities are statements that relate the size or order of two objects. The goal is to find all possible values of a variable that satisfy the inequality. For example, in the given exercise, you need to find the value of x that makes the inequality true.
Solving inequalities often involves similar steps to solving equations:
  • Isolate the variable on one side of the inequality.
  • Perform operations on both sides to maintain the inequality's direction.
Remember, if you multiply or divide both sides by a negative number, the inequality sign reverses.
Logarithmic Properties
Logarithms help us solve equations involving exponential functions. Here are some key properties of logarithms:
  • Product Rule: \(\text{log}(ab) = \text{log}(a) + \text{log}(b)\)
  • Quotient Rule: \(\text{log}(a/b) = \text{log}(a) - \text{log}(b)\)
  • Power Rule: \(\text{log}(a^b) = b \text{log}(a)\)
In the exercise, we apply the natural logarithm (ln) to both sides of the inequalities to utilize these properties. By doing so, we can manipulate the exponents and solve for x more easily.
Power Rule of Logarithms
The power rule of logarithms is especially helpful when dealing with exponential expressions. It states that the logarithm of an expression with an exponent can be written as the product of the exponent and the logarithm of the base:
  • For example, \(\text{log}(a^b) = b \text{log}(a)\)
In the given inequality \(3^{2x+1} \leq 7\), taking the logarithm of both sides allows us to bring down the exponent \(2x+1\) using the power rule. This results in a linear expression that we can solve for x.
Natural Logarithm
Natural logarithms (ln) have a special base e (approximately 2.718). They are particularly useful in calculus and in situations where growth or decay processes are involved.
In solving inequalities, the natural logarithm helps simplify exponential expressions. For instance, in \(0.8^x < 0.04\), applying the natural logarithm to both sides \(\text{ln}(0.8^x) < \text{ln}(0.04)\)\ brings the exponent x down. This makes it easier to isolate and solve for x.
Remember, the properties and rules that apply to regular logarithms also hold true for natural logarithms.

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Most popular questions from this chapter

During a recession a firm's revenue declines continuously so that the revenue, TR (measured in millions of dollars), in \(t\) years' time is modelled by $$ \mathrm{TR}=5 \mathrm{e}^{-0.15 \mathrm{t}} $$ (a) Calculate the current revenue and also the revenue in 2 years' time. (b) After how many years will the revenue decline to \(\$ 2.7\) million?

(Excel) Tabulate values of the total cost function $$ \mathrm{TC}=0.01 Q^{3}+0.5 Q^{2}+Q+1000 $$ when \(Q\) is \(0,2,4, \ldots, 30\) and hence plot a graph of this function on the range \(0 \leq Q \leq 30\). Use this graph to estimate the value of \(Q\) for which \(T C=1400\).

Show that the production function $$ Q=A\left[b K^{\alpha}+(1-b) L^{\alpha}\right]^{1 / \alpha} $$ is homogeneous and displays constant returns to scale.

Given that fixed costs are 100 and that variable costs are 2 per unit, express \(\mathrm{TC}\) and \(\mathrm{AC}\) as functions of \(Q\). Hence sketch their graphs.

(a) Use the power key \(x^{y}\) on your calculator to evaluate $$ \left(1+\frac{1}{m}\right)^{m} $$ where \(m=10000,100000\) and 1000000 . (b) Use your calculator to evaluate \(\mathrm{e}^{1}\) and compare with your answer to part (a).
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