Precalculus by Richard Wright

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3-02 Logarithmic Functions

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.4.1, PC.5.3

Trumpet player
Figure 1: credit (wikimedia/United States Marine Corps)

The trumpeter in the picture is testing his instrument before performing a ceremonial fanfare for former President Gerald Ford by Air Force One in Palm Springs, California. The trumpet player's loudness is about 80 dB while the airplane behind him is about 140 dB when it is taking off. Decibels operate on a logarithmic scale with a base of 10. That means the airplane's sound is 106 times as intense as the trumpet.

Logarithmic Functions

Lesson 3-01 dealt with exponential functions. To undo exponential functions, or to do the opposite of an exponential function, an inverse is needed. The inverse of an exponential function with base b is logarithmic function with base b.

Logarithmic Functions

y = logb x if and only if x = by

where x > 0, b > 0, and b ≠ 1.

A logarithmic function with base b is written as

f(x) = logb x

and read as

"log base b of x"

y = logb x

and

x = by

are equivalent. Notice that in the second equation, y is the exponent. That means that in the first equation, y is also an exponent. So logarithms are exponents.

Example 1: Rewrite a Logarithms and Exponentials

  1. Rewrite log3 243 = 5 as an exponential equation.
  2. Rewrite 26 = 64 as a logarithmic equation.
Solution
  1. Since logarithms equal exponents, 5 is the exponent. The base of the logarithm is also the base of the exponential, so 3 is the base. So, 35 = 243.
  2. Since logarithms equal exponents, the logarithm will equal 6. The base of the logarithm is the same as the base of the exponential, so the base is 2. So, log2 64 = 6.
Try It 1
  1. Rewrite log4 16 = 2 as an exponential equation.
  2. Rewrite 53 = 125 as a logarithmic equation.
Answers
  1. 42 = 16
  2. log5 125 = 3

To evaluate logarithms, you need to find what exponent of b will equal x.

Evaluate a Logarithm

If the logarithmic function is in the form

f(x) = logb x

Find which exponent on b equals x.

Example 2: Evaluate Logarithms

Evaluate

  1. log2 8
  2. log3 81
  3. log1⁄2 16
  4. log4 116
Solution
  1. Logarithms are exponents, so we need to rewrite the expression as an exponential. Since 23 = 8, the answer is 3.
  2. Since 34 = 81, the answer is 4.
  3. Since 1 2 - 4 = 16 , the answer is −4.
  4. Since 4 - 2 = 1 16 , the answer is −2.
Try It 2

Evaluate

  1. log5 25
  2. log218
Answers

a. 2; b. −3

Special Logarithms

There are two special logarithms that are commonly used in real-world applications. The first is the logarithm base 10 called the common log. The common log is written without the base like y = log x. The second is the logarithm base e called the natural log written as y = ln x. Both of these logarithms are on most scientific and graphing calculators. Most scientific and many graphing calculators only have these two logarithms on them and cannot calculate other logarithm bases without using properties that will be introduced in a later lesson.

Example 3: Evaluate Special Logarithms on a Calculator

Evaluate

  1. log 3
  2. log 10
  3. ln 1
  4. ln e3
Solution
  1. On most calculators type LOG 3 ENTER. It equals 0.477.
  2. Type LOG 10 ENTER. It equals 1. This makes sense because the base is 10 and logarithms equal the exponents. Since 101 = 10, the answer is 1.
  3. Type LN 1 ENTER. It equals 0. This is reasonable because the base is e, and e0 = 1. Since logarithms are the exponents, the answer is 0.
  4. Type LN ex 3 ENTER. It equals 3. This is because if it is written as an exponential, it is e3 = ey. So y = 3 and that is the answer.
Try It 3

Evaluate

  1. ln ½
  2. log 100
Answers

a. −0.693; b. 2

Properties of Logarithms

The preceding example demonstrates some properties of logarithms.

Properties of Logarithms
  1. logb 1 = 0 because b0 = 1.
  2. logb b = 1 because b1 = b.
  3. Inverse Property:
    1. logb bx = x because bx = bx.
    2. blogb x = x because rewritten as a logarithm this is logb x = logb x.
  4. One-to-One Property: If logb x = logb y, then x = y.

Example 4: Use Properties of Logarithms

Simplify

  1. log3 34
  2. ln 1
  3. eln 2
  4. log4 4
Solution
  1. Using property 3a, the answer is 4.
  2. Using property 1, the answer is 0.
  3. Using property 3b, the answer is 2.
  4. Using property 2, the answer is 1.
Try It 4

Simplify

  1. logπ 1
  2. 2log2 15
Answers

a. 0, b. 15

Applications

Decibels are units used to describe loudness of sound. For every 3 dB increase in decibels, the sound intensity is doubled. For every 10 dB increase in decibels, the sound loudness is doubled. The formula for calculating decibels is

β = 10  dB log I I 0

where I is the intensity, I0 is usually the threshold of hearing, or 10−12 W/m2, and β is the decibel level.

Example 5: Decibels

A loud rock concert can cause your ears to physically hurt and can cause permanent damage. If the intensity level is 1 W/m2, what is the decibel level of the rock concert?

Solution

I is 1 W/m2 and I0 is 10−12 W/m2. Fill these into the decibel formula and simplify.

β = 10  dB log 1 10 −12 = 10  dB log 10 12 = 10  dB 12 = 120 dB

A loud rock concert is about 120 dB and can cause pain and permanent hearing loss.

Lesson Summary

Logarithmic Functions

y = logb x if and only if x = by

where x > 0, b > 0, and b ≠ 1.

A logarithmic function with base b is written as

f(x) = logb x

and read as

"log base b of x"


Evaluate a Logarithm

If the logarithmic function is in the form

f(x) = logb x

Find which exponent on b equals x.


Properties of Logarithms
  1. logb 1 = 0 because b0 = 1.
  2. logb b = 1 because b1 = b.
  3. Inverse Property:
    1. logb bx = x because bx = bx.
    2. blogb x = x because rewritten as a logarithm this is logb x = logb x.
  4. One-to-One Property: If logb x = logb y, then x = y.

Helpful videos about this lesson.

Practice Exercises

  1. What does ln represent?
  2. Rewrite each equation in exponential form.

  3. logr (t) = w
  4. log6 (m) = n
  5. ln (7) = q
  6. Rewrite each equation in logarithmic form.

  7. ab = c
  8. p3 = 64
  9. et = 8
  10. Evaluate without using a calculator.

  11. \(\log_{2} \left(\sqrt{8}\right)\)
  12. \(4 \log_{3} \left(\frac{1}{9}\right)\)
  13. \(\ln \left(e^{\frac{2}{3}}\right)+4\)
  14. Use properties of logarithms to evaluate.

  15. eln(6.5) − 2
  16. 10log(16)
  17. Use a calculator to evaluate.

  18. log 15
  19. ln π
  20. Problem Solving

  21. How many decibels is a trumpet played with a sound intensity of 5×10−4 W/m2?
  22. Mixed Review

  23. (3-01) Evaluate 3 ⋅ 42.
  24. (3-01) State the domain, range, and asymptote of f(x) = 2 ⋅ ex + 1 − 3. Then graph the function.
  25. (2-07) Find the asymptotes and intercepts of g x = 2 x x - 1 .
  26. (2-06) Solve x3 + 2x2 − 9x = 18.
  27. (2-04) Divide (2x3 + x2 − 3) ÷ (x + 1).

Answers

  1. ln is a logarithm with base e called the natural log.
  2. rw = t
  3. 6n = m
  4. eq = 7
  5. loga c = b
  6. logp 64 = 3
  7. ln 8 = t
  8. \(\frac{3}{2}\)
  9. −8
  10. \(\frac{14}{3}\)
  11. 4.5
  12. 16
  13. 1.176
  14. 1.145
  15. 87.0 dB
  16. 48
  17. Domain: All Real Numbers; Range: (−3, ∞); HA: y = −3;
  18. VA: x = 1; HA: y = 2; x-int: (0, 0); y-int: (0, 0)
  19. −3, −2, 3
  20. 2 x 2 - x + 1 + - 4 x + 1