[PDF] PROPERTIES OF LOGARITHMIC FUNCTIONS





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PROPERTIES OF LOGARITHMIC FUNCTIONS

PROPERTIES OF LOGARITHMIC FUNCTIONS

EXPONENTIAL FUNCTIONS

An exponential function is a function of the form

()xbxf=, where b > 0 and x is any real number. (Note that ()2xxf= is NOT an exponential function.)

LOGARITHMIC FUNCTIONS

yxb=log means that ybx= where 1,0,0

¹>>bbx

Think: Raise b to the power of y to obtain x. y is the exponent. The key thing to remember about logarithms is that the logarithm is an exponent! The rules of exponents apply to these and make simplifying logarithms easier.

Example: 2100log10=, since 210100=.

x

10log is often written as just xlog , and is called the COMMON

logarithm. x elog is often written as xln, and is called the NATURAL logarithm (note: ...597182818284.2»e).

PROPERTIES OF LOGARITHMS

EXAMPLES

1. NMMNbbblogloglog+= 2100log2log50log

Think: Multiply two numbers with the same base, add the exponents. 2. NMN M bbblogloglog-= 18log756log7log56log8888==) Think: Divide two numbers with the same base, subtract the exponents.

3. MPMbP

bloglog= 623100log3100log3=×=×= Think: Raise an exponential expression to a power and multiply the exponents together. xbx b=log 01log=b (in exponential form, 10=b) 01ln

1log=bb 110log10= 1ln

=e xbx b=log xx=10log10 xex=ln xbx b=log Notice that we could substitute xyblog= into the expression on the left to form yb. Simply re-write the equation xyblog= in exponential form as ybx=. Therefore, xbbyx b==log. Ex: 2626ln=e

CHANGE OF BASE FORMULA

bNNaa blogloglog=, for any positive base a. 6476854.0079181.1698970.0

12log5log5log12»»=

This means you can use a regular scientific calculator to evaluate logs for any base. Practice Problems contributed by Sarah Leyden, typed solutions by Scott Fallstrom

Solve for x (do not use a calculator).

1. ()110log2 9=-x

2. 153log12

3=+x

3. 38log=x

4. 2log5=x

5. ()077log2

5=+-xx 6. 5.427log3=x

7. 2

38log-=x

8. ()11loglog66=-+xx 9. ()3loglog1

2221=+

xx 10. ()183loglog22

2=+-xx

11. ()()1loglog2 3

31321=-xx

Solve for x, use your calculator (if needed) for an approximation of x in decimal form.

12. 547=x

13. 17log

10=x

14. xx495×=

15. ex=10

16. 7.1=-xe

17. ()013.1lnln=x

18. xx98=

19. 4110ex=+

20. 54.110log-=x

Solutions to the Practice Problems on Logarithms:

1. ()1919109110log2212

9±=?=?-=?=-xxxx

2. 7142151233153log121512

3=?=?=+?=?=++xxxxx

3. 2838log3=?=?=xxx 4. 2552log2

5=?=?=xxx

5. ()()()1or 6160670775077log2202

5==?--=?+-=?+-=?=+-xxxxxxxxxx

6. ()5.15.435.43log5.43log5.427log3 33

33=?=?=?=?=xxxxx

7. 41
23

3223888log=?=?=?-=

--xxxx 8.

equation. original theosolution tonly theis 3 equation. new theonly solves which solution, extraneousan is 2 :Note .2or 30230661log11loglog222

666
x xxxxxxxxxxxxx 9. ( )641233

2222223log3log31loglog212121

2

1==?=?=?=))

---xxxxx xx 10. ( )( )2or 8028016616621log183loglog 22

8383222

222
xxxxxxxxxxxx xx 11.

729163332

3 31321

3331log1loglog1loglog

613
221
3 22
1 3

221==?=?=?=))

xxx xxxxxx

12. 0499.27log54log54log5477»=?=?=xxx

13. 17

101017log=?=xx

14. ()8467.99log99495 4545

45»?=?=?=?×=xxxxxxx

15. 4343.0loglog1010»=?=?=exexex

16. 5306.07.1ln7.1ln7.1-»-=?=-?=-xxex

17. ()7030.15ln013.1lnln

013.1013.1»=?=?=eexexx

18. ()01log198

8989=?=?=?=xxxxx

19. ()7372.0log10loglog1loglog1101044441

4»=?-=-=?=+?=+exxeexexe

20.quotesdbs_dbs2.pdfusesText_3
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