Logarithmic Functions Definition 1. For x > 0, a > 0, and a 1, y = log a x if and only if x = a y. The function f(x) = log a x is called the logarithmic function with base a. Example 1. Evaluate the following expressions. 1. log 2 16 2. log 7 = 1 49 Chapter 2: 2.3 Logarithmic Functions 1
The logarithmic function with base 10 is called the common logarithm and we commonly write this as log x, that is, if the base is omitted, it is understood to be 10. Another special logarithmic function is the natural logarithm with base e written as ln x. Chapter 2: 2.3 Logarithmic Functions 2
Example 2. Evaluate the following expressions. 1. log 1000 2. ln 1 e 2 Example 3. Sketch the graph of the following functions. 1. f(x) = log 2 x 2. f(x) = log 1/2 x Chapter 2: 2.3 Logarithmic Functions 3
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Properties of Logarithmic Functions f(x) = log a x, a > 0, a 1 1. The domain of f is the set of positive real numbers. 2. The range of f is the set of real numbers. 3. The x-intercept is at point (1, 0), that is, f(1) = 0. 4. f(a) = 1. 5. If 0 < a < 1, as the value of x increases, the value of y decreases. 6. If a > 1, as the value of x increases, the value of y increases. 7. The function f is one-to-one, that is, log a x = log a y if and only if x = y. 8. The graph of f has a vertical asymptote x = 0 or the y-axis. Chapter 2: 2.3 Logarithmic Functions 5
Laws of Logarithms Let x, y and r be real numbers, and a a positive number other than 1. 1. log a xy = log a x + log a y 2. log a x y = log a x log a y 3. log a x r = r log a x Chapter 2: 2.3 Logarithmic Functions 6
Example 4. Solve each equation. 1. log x 64 1/3 = 1 2 2. log 4 (2x + 11) = log 4 (5x 4) 3. log(x + 3) + log x = 1 Example 5. Solve 2 3x 2 = 5. Round off answer to four decimal places. Chapter 2: 2.3 Logarithmic Functions 7
Solving Logarithmic Inequalities 1. Write the inequality in the standard form. One side must be zero and the other side is a single logarithmic expression which we denote by f(x). 2. Find the critical numbers. These are the zeros of f(x). 3. Determine the domain restriction. 4. Set up the intervals on the number line using the critical numbers obtained in Step 2 and the domain restriction obtained in Step 3. 5. Choose a test value in each interval in Step 4, and construct a table. Substitute the test value to f(x) and determine the sign of the resulting answer. The sign of this answer (positive or negative) will be the sign of the entire interval. You can check using a different number from the same interval if you want to verify your answer. Chapter 2: 2.3 Logarithmic Functions 8
6. Use the table in step 5 to determine which intervals satisfy the inequality. If the inequality is of the form f(x) < 0 or f(x) 0 then all of the intervals with the negative sign are solutions. In addition, the zeros of f(x) are part of the solution if f(x) 0. On the other hand, if the inequality is of the form f(x) > 0 or f(x) 0 then all of the intervals with the positive sign are solutions. In addition, the zeros of f(x) are part of the solution if f(x) 0. Chapter 2: 2.3 Logarithmic Functions 9
Example 6. Solve 2 log 49 (2x + 1) 1. Example 7. Solve log(x 3) > log(x + 6) 1. Chapter 2: 2.3 Logarithmic Functions 10
The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. Example 8. Use the properties of logarithms to expand each expression. 1. log 6 x 2 y 3 z 2. ln 5x 3 4 Example 9. Use the properties of logarithms to condense each logarithmic expression. 1. 1 log x + 3 log(x + 3) 3 3. 1 2 [log 2 x + log 2 (x + 1)] 2. 2 ln(x 2) ln x1 Chapter 2: 2.3 Logarithmic Functions 11
Example 10. If log 3 5 = 1.46 and log 3 7 = 1.77, use logarithmic properties to find ( ) 7 2. log 3 45 1. log 3 5 Example 11. Find x so that log b x = 2 3 log b 8 + 1 2 log b 9 log b 6 without using a calculator. Chapter 2: 2.3 Logarithmic Functions 12
Example 12. The bacterium Escherichia coli (E. coli) is found naturally in the intestines of many mammals. In a particular laboratory experiment, the doubling time for E. coli is found to be 30 minutes. If the experiment starts with a population of 1,000 E. coli and there is no change in the doubling time, how long would it take to have a population of 5,000 E. coli? Example 13. You discovered a new radioactive isotope. It s half life is 1.23 years. How long would it take for a sample of 45 grams, to decay to 5 grams? Example 14. If P10,000 is invested in an account paying 10% compounded monthly, how long will it take for the money to double? Example 15. If P10,000 is invested for 10 years, how much should the interest rate be for the money to double? Assume that the bank offers continuous compounding. Chapter 2: 2.3 Logarithmic Functions 13
Magnitudes of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensity I is given by R = log I I 0 where I 0 = 1 is the minimum intensity used for comparison. Chapter 2: 2.3 Logarithmic Functions 14
Example 16. In October 15, 2013, Bohol experienced an earthquake that measured 7.2 on the Richter scale while in March 30, 2015 it experienced a 4.7 magnitude. Find the intensity of each earthquake and compare the two intensities. Chapter 2: 2.3 Logarithmic Functions 15
Exercises Evaluate the following expressions. 1. log 2 1 16 2. log 1 6 1 4 3. log 0.01 4. log 49 343 5. ln e 6. log 8 16 7. log 8 4096 Chapter 2: 2.3 Logarithmic Functions 16
g. Sketch the graph of f. Then use the graph of f to sketch the graph of 1. f(x) = 3 x, g(x) = log 3 x 2. f(x) = 5 x, g(x) = log 5 x 3. f(x) = e 2x, g(x) = 1 2 ln x Chapter 2: 2.3 Logarithmic Functions 17
Solve each equation. 1. log x log 5 = log 2 log(x 3) 7. log 7 4x log 7(x + 1) = 1 2 log 7 4 2. log(6x + 5) log 3 = log 2 log x 8. 2 log 3 x = log 3 2 + log 3(4 x) 3. log(x 2 2x 2) = 2 log(x 2) 4. ln x = ln(2x 1) ln(x 2) 5. ln(x + 1) = ln(3x + 1) ln x 6. ln(x + 8) ln x = 3 ln 2 9. 3 log b 2 + 1 2 log b 25 log b 20 = log b x 10. 3 2 log b 4 2 3 log b 8+2 log b 2 = log b x Chapter 2: 2.3 Logarithmic Functions 18
Solve the exponential equation in terms of natural or common logarithms. 1. 3 x = 25 2. 5 2x 3 = 18 3. 4 + 3 x+1 = 8 4. e 2x 5 = 7 5. 2 x 7 = 3 x+5 Chapter 2: 2.3 Logarithmic Functions 19
Solve each inequality. 1. log 6 (x + 10) 3 2. 2 log x < log(2x 1) 3. log x + log(2 x) < 1 4. log(x + 2) < 2 log(2x 6) 5. log 1 (3x 1) log 1 + 2) > 1 3 3(x Chapter 2: 2.3 Logarithmic Functions 20
Use the properties of logarithms to expand each expression. 1. log 3 9a 2 bc 3 2. log 8 x 14 y 7 3. ln z 15 x x2 + 1 4. log b xy 4 z 4 5. log 9 6 x 5 y 3 (z 2) 7 Chapter 2: 2.3 Logarithmic Functions 21
Use the properties of logarithms to condense each logarithmic expression. 1. 1 3 log b x log b y 5 log b z 2. ln x 2[ln(x + 2) + ln(x 2)] 3. 4[log z + log(z + 5)] 2 log(z 5) 4. 1 2 [log 3(x + 3) + log 3 x log 3 (x 2 1)] 5. 3[log a x log a (x + 1) log a (x 1)] Chapter 2: 2.3 Logarithmic Functions 22
Answer the following questions. 1. An investment of P100,000 grew to P125,000. If interest was computed at 12% compounded monthly, how long was the money invested? 2. At what annual rate compounded continuously will P100,000 have to be invested to amount to P250,00 in 10 years? 3. After t years, the annual sales in hundreds of thousands of units of a product q is given by ( ) 0.6t 1 q =. 2 After how many years will the annual sales about 100,000 units? 4. The loudness of a sound is often given on the decibel (db) scale. The loudness of a sound L in db is defined in terms of its intensity I by the Chapter 2: 2.3 Logarithmic Functions 23
equation L = 10 log I I 0 where I 0 is the minimum intensity audible to the average person. (I0 = 1.0 10 12 W/m 2. (a) A typical rock concert can have an intensity of 1W/m 2. What is the loudness in decibels of such a concert? (b) A whisper has a loudness of 20 db. What is the intensity of the whisper? 5. The sales S (in thousands of units) of a cleaning solution after x thousand pesos is spent on advertising are given by S = 10(1 e kx ). When P5,000 is spent on advertising, 2500 units are sold. (a) Complete the model by solving for k. (b) Estimate the number of units that will be sold if advertising expenditures are raised to P7,000. Chapter 2: 2.3 Logarithmic Functions 24