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Module 1: Basic Concepts - Logarithms
Logarithms to Base TenNatural Logarithms
Practice Problems
Objective
- Solve equations involving logarithms in base ten and natural logarithms.
A logarithm is a way of expressing numbers as a function of a base number such as 10. For example, two is the logarithm of 100 in base ten. This logarithmic function and its equivalent exponential form are shown in the three examples below. It is apparent that the logarithmic function is the inverse of the exponential function.
The logarithmic function provides a way to express a very large number in a very concise way. For example, refrigeration and/or cryogenic condensers can be added to the air pollution control system to reduce gaseous emissions. As the condenser cools the gas stream, the temperature and the vapor pressure of the gas decrease. The vapor pressure of organic compounds in the gas stream can vary from values as low as 0.001 mm Hg to more than 100 mm Hg, depending on the operating temperature in the condenser. Thus, the vapor pressure can span a 100,000 mm Hg range. The two graphs in Figure 1 plot the same data in two different ways: with and without logarithms. It is difficult to express the vapor pressure at low gas temperatures without the use of logarithms. Logarithms allow you to convert exponential data to a format that can easily fit on a graph.
Using logarithms is also helpful in the field of air pollution when important operating conditions vary over a wide range. For example, the pH scale is a logarithmic way to express H+ (sometimes written H3O+) ion concentration over a 14 order of magnitude scale.
The laws of handling logarithms are similar to those for handling exponents. Some examples are illustrated below.
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In addition to the base of ten, the natural (sometimes termed "Napierian") base is often used in air pollution-related work. The natural base is the number 2.718282. The number is often expressed as the letter "e." For example, the expression e2 is equivalent to (2.718282)2. When natural logs are used, the symbol is written as Ln rather than Log2.718282, as shown below.
Practice Problems
Logarithms
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Instructions:
- Complete the Practice Problems before proceeding to the next lesson. Click on the button below.