CHAPTER 3 SUPPLEMENTARY EXERCISES
4.2 LOGARITHMIC FUNCTIONS
LOGARITHMIC FUNCTIONS 113
5. a) 105x
55x b) 322x
162x c) 43x
123x 6. a) 203x
53x b) 187x
67x c) 6x
24x In Exercises 7–12, simplify the expressions using the laws of exponents.
7. 7x3x5y−2 x2y4 8.
(2x3 y4
)−2
9. x3 y−2 ÷ x
y5
10.
(5a3b4c0 10abc2
)−3
11. 25x+34x+1 8(23x−1) 12. 94x−127x+2
81x+3 In Exercises 13–20, solve for x.
13. 73x=715
14. 10−3x=1,000,000 15. 27−x=32
16. 32x3x+1 =81
17. (1+x)5x+ (3−2x)5x=0 18. (x2)4x−9(4x) =0
19. (x2+4x)7x+ (x+6)7x=0 20. (x3)5x− (7x2)5x+ (12x)5x=0 In Exercises 21–25, the expressions can be factored. Find the missing factor.
21. 23+h =2h( ) 22. 35+h =35( ) 23. 7x+5−72x=72x( )
24. 52h−9= (5h−3)( ) 25. 73h−8= (7h−2)( )
–8 –6 –4 –2 0 2 4 6
0 2 4 6 8 10
y
x log2x = y
While the domain forxis(0,∞), the range foryis all real numbers. Thex-intercept is atx=1. There is noy-intercept.
Example 4.2.1 Exponential and Logarithmic Forms Convert exponentials to logarithms or vice versa.
a)34=81 b)211=2048 c)log41024=5 d)log21∕8= −3 Solution:
a) The exponential is converted to the logarithm log381=4.
b) The exponential is converted to the logarithm log22048=11.
c) The logarithm is converted to the exponential 45 =1024.
d) The logarithm is converted to the exponential 2−3=1∕8.
Most calculators have two logarithmic keys: logx and lnx. The logx has a base 10 (log10x)and lnxhas a basee(logex). Base 10 logarithms are calledcommon logarithms, while baseelogarithms are callednatural logarithms. Note that the base numerals 10 andeare identified by the symbolic spellinglogandln, respectively.
↑ Recall, any positive number can be an exponential base. For instance, log28=3 as 23=8. Likewise, log636=2 as 62 =36.
Useful properties of common logarithms and natural logarithms for algebraic functions, f(x) andg(x), are:
General Properties of Logarithms
logb1=0 logb(f(x)g(x)) =logbf(x) +logbg(x) logbb=1 logb(f(x)∕g(x)) =logbf(x) −logbg(x) logbbx=x logb[f(x)r] =rlogbf(x)
blogbx =x
LOGARITHMIC FUNCTIONS 115
Natural Logarithms
ln 1=0 lnf(x)g(x) =lnf(x) +lng(x) lne=1 ln(f(x)∕g(x)) =lnf(x) −lng(x) lnex=x ln[f(x)r] =rlnf(x)
elnx=x
A logarithm is inreducedor simplest form when component products, quotients, or exponents are irreducible. It takes practice to become thoroughly familiar with these prop- erties of exponents and logarithms.
Example 4.2.2 Simplifying Logarithms
Express these logarithms in reduced form.
a)log3(x+1)(3x−5) b)log7
((x+2)3(x−4) (x−5)2
)
c)ln(x2+2)3(2y−1)4(z+3)2 Solution:
a) In this case, the logarithm is written as a sum of logarithms, as log3(x+1) +log3(3x−5)
b) Firstly, decompose as log7(x+2)3+log7(x−4) −log7(x−5)2.
Next, place the exponents as coefficients: 3log7(x+2) +log7(x−4) −2log7(x−5).
The absence of products, quotients, or exponents completes the simplification.
c) Firstly, expand the logarithms as ln (x2+2)3+ln(2y−1)4+ln(z+3)2. Then, using exponents as coefficients, 3ln(x2+2) +4ln(2y−1) +2ln(z+3). Again, a simplest form.
⧫John Napier discovered logarithms in the early seventeenth century. They are sug- gested by the exponents of 10,100,101,102,103, …; the base of the decimal system.
Any positive number can be a logarithmic base. Later discoveries led to a preference for the “log naturalis” or natural logarithms that have the basee.
Logarithms were especially important for arithmetic computation before modern cal- culators and computers. They formed the basis for the slide rule; every engineering student’s companion until the 1970s. Logarithms often appear in mathematical models of natural phenomena. (Historical Notes)
Example 4.2.3 Combining Logarithms
Express as a single logarithm:
a) log3(x+1) +2log3(x+4) −7log3(2x+5) b) 5ln(x+3) −3ln(y+2) −2ln(z+5)
Solution:
a) As bases are the same, use the coefficients as exponents to yield:
log3(x+1) +log3(x+4)2 −log3(2x+5)7 Next, a sum (difference) of logarithms becomes a product (quotient) as log3(x+1)(x+4)2
(2x+5)7 .
b) Rewriting, coefficients become exponents as:
ln(x+3)5 −ln(y+2)3 −ln(z+5)2. The two terms with negative coefficients form the denominator as
ln (x+3)5 (y+2)3(z+5)2.
⧫The widely quoted Richter scale is one example of a widely used base 10 logarithmic measure. It was devised by Charles Richter in 1935 to compare earthquake magnitudes.
As you know from news reports, on the Richter scale, an earthquake of, say, magnitude 5 is ten times a magnitude 4, and so on.
A major earthquake has magnitude 7, while a magnitude 8 or larger is called a Great Quake.
A Great Quake can destroy an entire community. The Indian Ocean Quake of December 2004, magnitude 9.0 on the Richter scale, resulted in Tsunamis that caused massive destruction.
The Great Chilean Earthquake of 1960, magnitude 9.5 on the Richter scale, is the strongest earthquake on record.
Other logarithmic scales include the decibel in acoustics, the octave in music, f-stops in photographic exposure, and entropy in thermodynamics. The pH scale chemists use to measure acidity and the stellar magnitude scale used by astronomers to measure the star brightness are also examples of logarithmic scales.
LOGARITHMIC FUNCTIONS 117 Logarithms and exponentials, being inverses, sometimes enable one to solve exponential equations as in the following examples.
Example 4.2.4 Solving an Exponential Equation (Base 10) Solve using logarithms.
a) 10x=9.95 b) 102x+1=1050
Solution:
a) As 100 =1 and 101=10, it follows that 0<x<1. Note that x is close to 1 as 9.95 is close to 10. The logarithm of the equation (base 10, here) yields:
log1010x=log109.95 and, using a calculator, x=log109.95≈0.9978
in agreement with our estimate.
b) As 103=1000 and 104=10,000, an estimate of x can be obtained from
3<2x+1 < 4. This yields 1<x<1.5. Taking the logarithm of the equation (base 10, here) yields:
log10102x+1=log101050 so 2x+1=log101050
x= −1+log101050
2 ≈1.0106
we could have anticipated that x≈1 without calculation as x=1 yields 103=1000≈1050.
Example 4.2.5 Determining an Exponent (Base e)
Solve using properties of logarithms
5e4x−1=100 Solution:
Note that e4x−1 =20. Using a calculator, e3≈20. Therefore, 4x−1≈3, so x is close to 1. Taking the natural logarithm of both sides (base e, here) yields:
lne4x−1 =ln20 4x−1=ln20
x= 1+ln 20
4 ≈0.9989 The value for x agrees with our preliminary estimate.
EXERCISES 4.2
1. Graphy=log3x 2. Graphy=lnx In Exercises 3–12, evaluate the logarithms.
3. log101,000,000 4. log3243 5. log264 6. log5125 7. log2 1
32
8. log3 1 81 9. lne3 10. lne7 11. lne7.65 12. lne−3.4 In Exercises 13–20, evaluate the expressions.
13. ln(lne) 14. eln 1 15. log927 16. log25125
17. log432 18. log5625 19. log2128 20. log4(1∕64) In Exercises 21–36, solve for x.
21. logx27=3 22. logx64=3 23. log3(5x+2) =3 24. log2(x2+7x) =3
25. ln 5x=ln 35 26. e3xe2x=2 27. ln(ln 4x) =0 28. ln(7−x) =1∕3 29. Write in reduced form: log4(x+1)2(x−3)6
(3x+5)3 . 30. Write in reduced form: ln (x−1)4
(2x+3)5(x−4)2.
31. Write as a single logarithm: 2 lnx−3 ln(y+1) +4 ln(z+1).
DERIVATIVES OF EXPONENTIAL FUNCTIONS 119 32. Write as a single logarithm: ln 2−ln 3+ln 7.
33. 102x−1=105 34. 103x−1=100,100 35. 3ex−1 =4 36. e3x+1 =22
37. In June 2004, an earthquake of magnitude 4.1 struck Northern Illinois. Its effects were felt from Wisconsin to Missouri and from western Michigan to Iowa. Another earth- quake in the Midwest, the 1895 Halloween Earthquake, is estimated at 6.8 on the Richter scale. How much stronger was the Halloween Quake than the Northern Illinois Quake?