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Chapter 13: Problem 35

Find the indicated values. Find \(e\) if \(\log e=-0.35,\) where \(e\) is the efficiency of a certain gasoline engine.

Short Answer

Expert verified
The efficiency \(e\) is approximately 0.4467.

Step by step solution

01

Understanding the Logarithmic Equation

The problem provides the equation \(\log e = -0.35\). Here, we need to find the value of \(e\), which is inside a logarithmic function.
02

Using the Inverse of Logarithm

To find \(e\), we need to remove the logarithm. Since the logarithm with base 10 is used here (common logarithm), we use the inverse operation, which is raising 10 to the power of the value. Thus, we write it as \(e = 10^{-0.35}\).
03

Calculating the Power

Now we calculate \(10^{-0.35}\). This represents taking the reciprocal of \(10^{0.35}\). Using a calculator, evaluate \(10^{0.35}\), which equals approximately 2.2387. Therefore, \(e = \frac{1}{2.2387}\).
04

Final Calculation

Divide 1 by 2.2387 to get the final efficiency value. Doing this, \(e \approx 0.4467\).

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Key Concepts

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

Efficiency Calculation
In engineering, efficiency calculations help evaluate the performance of energy systems, like gasoline engines. Efficiency is the ratio of the output energy to the input energy. Calculating it accurately can reflect how well a system converts energy to useful work, which is crucial for understanding improvements and sustainability.
  • Efficiency (\( e \)) can often be represented in different forms, such as percentages. In our example, it's a decimal between 0 and 1.
  • The value informs engineers how much of the input fuel energy is converted into useful movement or work.
  • Understanding efficiency is essential for improving engine designs and reducing waste.
Whether it involves thermodynamics or simple mechanical systems, knowing how to calculate efficiency can drastically impact energy use and economic factors.
Common Logarithms
Common logarithms, denoted by \( \log \), have a base of 10. They are heavily used because log scales can simplify calculations across engineering, science, and finance.
  • This logarithmic scale helps manage large numbers or fractions that occur frequently in scientific measures.
  • In the problem, \( \log e = -0.35 \) means that \( e \) represents some efficiency calculation.
  • The negative logarithmic value indicates that \( e \) is a fraction between 0 and 1.
Moreover, common logarithms convert multiplicative processes into additive ones, aiding easier handling and interpretation in analysis and reporting.
Inverse Operations
Inverse operations allow us to undo algebraic functions. For logarithms, the inverse is the exponential function. Here's how it works:
  • To solve \( \log e = -0.35 \), we need the inverse operation: raising 10 to the power of the other side of the equation.
  • This process involves converting a logarithmic equation back to its original exponential form: \( e = 10^{-0.35} \).
  • Applying inverse operations reveal the original value on which the logarithm operated.
Inverse operations are pivotal for manipulating equations in math and science, allowing transformation of complex formulas into solvable ones. Understanding this concept can help solve various mathematical challenges efficiently.

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Most popular questions from this chapter

Use a calculator to solve the given equations. In chemistry, the pH value of a solution is a measure of its acidity. The pH value is defined by \(\mathrm{pH}=-\log \left(\mathrm{H}^{+}\right),\) where \(\mathrm{H}^{+}\) is the hydrogen-ion concentration. If the pH of a sample of rainwater is \(4.764,\) find the hydrogen-ion concentration. (If \(\mathrm{pH}<7,\) the solution is acid. If \(\mathrm{pH}>7,\) the solution is basic.) Acid rain has a pH between 4 and \(5,\) and normal rain is slightly acidic with a pH of about 5.6

$$\text {Plot the indicated graphs.}$$ The magnetic intensity \(H\) (in \(\mathrm{A} / \mathrm{m}\) ) and flux density \(B\) (in teslas) of annealed iron are given in the following table. Plot \(H\) as a function of \(B\) on log-log paper. (The unit tesla is named in honor of Nikola Tesla. See Chapter 20 introduction.) $$\begin{aligned} &\begin{array}{l|c|c|c|c} B(\mathrm{T}) & 0.0042 & 0.043 & 0.67 & 1.01 \\ \hline H(\mathrm{A} / \mathrm{m}) & 10 & 50 & 100 & 150 \end{array}\\\ &\begin{array}{l|l|l|l|l} B(\mathrm{T}) & 1.18 & 1.44 & 1.58 & 1.72 \\ \hline H(\mathrm{A} / \mathrm{m}) & 200 & 500 & 1000 & 10,000 \end{array} \end{aligned}$$

If interest is compounded continuously (daily compounded interest closely approximates this), with an interest rate \(i,\) a bank account will double in \(t\) years according to \(i=(\ln 2) / t .\) Find \(i\) if the account is to double in 8.5 years.

$$\text {Plot the indicated graphs.}$$ The acceleration \(g\) (in \(\mathrm{m} / \mathrm{s}^{2}\) ) produced by the gravitational force of Earth on a spacecraft is given by \(g=3.99 \times 10^{14} / r^{2},\) where \(r\) is the distance from the center of Earth to the spacecraft. On log-log paper, graph \(g\) as a function of \(r\) from \(r=6.37 \times 10^{6} \mathrm{m}\) (Earth's surface) to \(r=3.91 \times 10^{8} \mathrm{m}\) (the distance to the moon).

If \(\ln x=3\) and \(\ln y=4,\) find \(\sqrt{x^{2} y}\).
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