Properties of Logarithms

PROPERTIES OF LOGARITHMIC FUNCTIONS

EXPONENTIAL FUNCTIONS

An exponential function is a function of the form f (x) = b x , where b > 0 and x is any real number. (Note that f (x) = x2 is NOT an exponential function.)

LOGARITHMIC FUNCTIONS

logb x = y means that x = b y where x > 0,b > 0, b 1

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: log10 100 = 2 , since 100 = 102 .

log10 x is often written as just log x , and is called the COMMON logarithm. loge x is often written as ln x , and is called the NATURAL logarithm (note: e 2.718281828459...).

PROPERTIES OF LOGARITHMS

EXAMPLES

1. logb MN = log b M + logb N

log 50 + log 2 = log100 = 2

Think: Multiply two numbers with the same base, add the exponents.

2.

log b

M N

= log b M - logb N

log

8

56

-

log

8

7

=

log

8

56 7

=

log

8

8

=

1

Think: Divide two numbers with the same base, subtract the exponents.

3. logb M P = P logb M

log1003 = 3 log100 = 3 2 = 6

Think: Raise an exponential expression to a power and multiply the exponents together.

logb b x = x logb b = 1 logb b x = x b logb x = x

logb 1 = 0 (in exponential form, b0 = 1)

ln1 = 0

log10 10 = 1

ln e = 1

log10 10 x = x

ln e x = x

Notice that we could substitute y = logb x into the expression on the left

to form b y . Simply re-write the equation y = log b x in exponential form

as x = b y . Therefore, blogb x = b y = x .

Ex: eln 26 = 26

CHANGE OF BASE FORMULA

log b

N

=

log a N log a b

,

for

any

positive

base

a.

log12

5

=

log 5 log12

0.698970 1.079181

0.6476854

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

( ) 1. log9 x2 -10 = 1

Solve for x (do not use a calculator).

6. log3 27 x = 4.5

10. log 2 x2 - log 2 (3x + 8) = 1

2. log3 32x+1 = 15 3. log x 8 = 3 4. log5 x = 2

( ) 5. log5 x2 - 7x + 7 = 0

7.

log

x

8

=

-

3 2

8. log 6 x + log 6 (x -1) = 1

( ) 9.

log 2

x 1 2

+ log 2

1 x

=3

11.

(

1 2

)log

3

x

-

(

1 3

)log

3

x2

=1

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

12. 7 x = 54

15. 10 x = e

18. 8x = 9 x

13. log10 x = 17 14. 5x = 9 4 x

16. e-x = 1.7

17. ln(ln x) = 1.013

19. 10 x+1 = e4 20. log x 10 = -1.54

Solutions to the Practice Problems on Logarithms:

( ) 1. log9 x 2 -10 = 1 91 = x2 -10 x 2 = 19 x = ? 19

2. log3 32x+1 = 15 315 = 32x+1 2x + 1 = 15 2x = 14 x = 7

3. log x 8 = 3 x3 = 8 x = 2

4. log5 x = 2 52 = x x = 25

( ) 5. log5 x2 - 7x + 7 = 0 50 = x2 - 7x + 7 0 = x2 - 7x + 6 0 = (x - 6)(x -1) x = 6 or x = 1

( ) 6. log3 27 x = 4.5 log 3 33 x = 4.5 log 3 33x = 4.5 3x = 4.5 x = 1.5

7.

log x 8 =

-

3 2

x -32

=8

x

= 8-23

x

=

1 4

( ) log6 x + log6 (x -1) = 1 log6 x2 - x = 1 x2 - x = 6 x2 - x - 6 = 0

8. (x - 3)(x + 2) = 0 x = 3 or x = -2. Note : x = -2 is an extraneous solution, which solves only

the new equation. x = 3 is the only solution to the original equation.

( ) 9.

log 2

x 1 2

+

log

2

1 x

=

3

log 2

x1 2 x

=

3

log 2

x -12

= 3 23

=

x-

1 2

x=

23

-2

=

1 64

( ) ( ) 10. log2 x2 - log 2 3x + 8

= 1 log 2

x 2 3x+8

=1

x2 3x+8

=

2

x2

=

6x + 16

x2 - 6x -16 = 0 (x - 8)(x + 2) = 0 x = 8 or x = -2

( ) ( ) 11.

1 2

log3 x -

1 3

log3

x2

=1

log3

x1 2

- log3

x2 3

=1

log 3

x1 2

x2 3

=1

x 12-23

=3

x-

1 6

=

3

x

=

3-6

=

1 729

12.

7x

=

54

x

=

log 7 54

x

=

log 54 log 7

2.0499

13. log10 x = 17 x = 1017

( ) 14.

5x

= 94x

5x 4x

=9

5 4

x = 9 x = log 5 9 x 9.8467

4

15. 10 x = e x = log10 e x = log e 0.4343

16. e-x = 1.7 -x = ln 1.7 x = - ln1.7 -0.5306

17. ln(ln x) = 1.013 ln x = e1.013 x = ee1.013 15.7030

18.

8x

= 9x

1=

( )9 x 8

x

=

log 9 1

x

=

0

8

( ) 19.

10 x+1

=

e 4

x

+1=

log e4

x

=

log e 4

-1=

log e 4

- log10

x

=

log

e4 10

0.7372

20.

log x 10

=

-1.54

x -1.54

= 10

x

= 10-

1 1.54

0.2242

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