14. 1 Exponential Functions and Applications 14 14.1 Exponential Functions and Applications 14.3 Logarithmic Functions and Applications

14.3 Logarithmic Functions

Logarithms have many applications inside and outside mathematics Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor

•To find the number of payments on a loan or the time to reach an investment goal •To model many natural processes, particularly in living systems. We perceive loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale. We also perceive brightness of light as the logarithm of the actual light energy, and star magnitudes are measured on a logarithmic scale. •To measure the pH or acidity of a chemical solution. The pH is the negative logarithm of the concentration of free hydrogen ions. •To measure earthquake intensity on the Richter scale. •To analyze exponential processes. Because the log function is the inverse of the exponential function, we often analyze an exponential curve by means of logarithms. Plotting a set of measured points on “log-log” or “semi-log” paper can reveal such relationships easily. Applications include cooling of a dead body, growth of bacteria, and decay of a radioactive isotopes. The spread of an epidemic in a population often follows a modified logarithmic curve called a “logistic”. •To solve some forms of area problems in calculus. (The area under the curve 1/x, between x=1 and x=A, equals ln A.)

Other uses that are more aligned with the scope of this course are determining decibel levels for sound (loudness) and severity of earthquakes. Logarithms are used in other areas but the math needed to apply them is advanced beyond this class. We will use a few of these concepts to learn how to deal with logs through that use hopefully you learn an appreciation for the need of logs.

In its simplest form, a logarithm answers the question: First of all, what is a logarithm? You’ve worked with them before, you’ve learned the rules before, but do you really know what one is? In its simplest form, a logarithm answers the question: How many of one number do we multiply to get another number? For example how many 2s do you need to multiply together to get 8? The log then would be 3. The notation would be How many of the bases “2” do you need to multiply by itself to get 8?

Remember logs and exponentials are inverses of one another. Take your calculator, find the log button, the log button on your calculator uses a base of 10. Later we can change that, but for right now our calculator is only useful when the base is 10. Verify the above value with your calculator. What does this mean. To get a value of 6.3, you need to multiply 10 by itself 0.8 times.

Common Log A log that involves a base of 10 is referred to as the common log. The definition is as follows ….

General Log Rule for logs of bases other than 10

Convert these logs into exponential form

Write the following in log form 103 = 1,000 42 = 16 33 = 27 51 = 5 70 = 1 4-2 = 1/16 251/2 = 5

Examples of Logarithms Exponential Form Logarithmic Form Example Solve Solution

Solving Logarithmic Equations Example Solve a) Solution Since the base must be positive, x = 2.

The Common Logarithm – Base 10 Example Evaluate Solution Use a calculator. For all positive numbers x,

The Natural Logarithm – Base e On the calculator, the natural logarithm key is usually found in conjunction with the e x key. For all positive numbers x,