Properties of Logarithms - Shoreline Community College

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 ) = x 2 is NOT an exponential function.)

LOGARITHMIC FUNCTIONS

log b x = y means that x = b y where x > 0, b > 0, b ¡Ù 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: log 10 100 = 2 , since 100 = 10 2 .

log 10 x is often written as just log x , and is called the COMMON logarithm.

log e x is often written as ln x , and is called the NATURAL logarithm (note: e ¡Ö 2.718281828459... ).

PROPERTIES OF LOGARITHMS

EXAMPLES

1. log b MN = log b M + log b N

log 50 + log 2 = log 100 = 2

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

M

? 56 ?

= log b M ? log b N

log 8 56 ? log 8 7 = log 8 ? ? = log 8 8 = 1

N

? 7?

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

2. log b

3. log b M P = P log b M

log 100 3 = 3 ? log 100 = 3 ? 2 = 6

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

log b b x = x

log b 1 = 0 (in exponential form, b 0 = 1 )

log b b = 1

log 10 10 = 1

ln e = 1

log b b = x

log 10 10 = x

ln e x = x

b logb x = x

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

x

x

ln 1 = 0

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

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

Ex: e ln 26 = 26

CHANGE OF BASE FORMULA

log b N =

log a N

, for any positive base a.

log a b

log 12 5 =

log 5 0.698970

¡Ö

¡Ö 0.6476854

log 12 1.079181

This means you can use a regular scientific calculator to evaluate logs for any base.

1.

2.

3.

4.

Practice Problems contributed by Sarah Leyden, typed solutions by Scott Fallstrom

Solve for x (do not use a calculator).

2

log 9 (x ? 10 ) = 1

6. log 3 27 x = 4.5

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

3

log 3 3 2 x +1 = 15

7. log x 8 = ?

11. ( 1 2 ) log 3 x ? ( 13 ) log 3 x 2 = 1

2

log x 8 = 3

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

log 5 x = 2

1

9. log 2 x 2 + log 2 ( 1x ) = 3

log (x 2 ? 7 x + 7 ) = 0

5.

5

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. 8 x = 9 x

13. log 10 x = 17

16. e ? x = 1.7

19. 10 x +1 = e 4

17. ln (ln x ) = 1.013

20. log x 10 = ?1.54

14. 5 x = 9 ? 4 x

Solutions to the Practice Problems on Logarithms:

1. log 9 (x 2 ? 10) = 1 ? 91 = x 2 ? 10 ? x 2 = 19 ? x = ¡À 19

2. log 3 3 2 x +1 = 15 ? 315 = 3 2 x +1 ? 2 x + 1 = 15 ? 2 x = 14 ? x = 7

3. log x 8 = 3 ? x 3 = 8 ? x = 2

4. log 5 x = 2 ? 5 2 = x ? x = 25

5. log 5 (x 2 ? 7 x + 7 ) = 0 ? 5 0 = x 2 ? 7 x + 7 ? 0 = x 2 ? 7 x + 6 ? 0 = ( x ? 6)( x ? 1) ? x = 6 or x = 1

( )

6. log 3 27 x = 4.5 ? log 3 33

x

= 4.5 ? log 3 33 x = 4.5 ? 3 x = 4.5 ? x = 1.5

7. log x 8 = ? 32 ? x ? 2 = 8 ? x = 8 ? 3 ? x =

3

2

1

4

log 6 x + log 6 ( x ? 1) = 1 ? log 6 (x 2 ? x ) = 1 ? x 2 ? x = 6 ? x 2 ? 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 t o the original equation.

?x 2

?1?

9. log 2 x + log 2 ? ? = 3 ? log 2 ??

? x?

? x

?

1

1

? = 3 ? log 2 x ? 2 = 3 ? 2 3 = x ? 2 ? x = 2 3

?

?

2

2

x2

log 2 x ? log 2 (3 x + 8) = 1 ? log 2 3 x +8 = 1 ? 3 xx+8 = 2 ? x 2 = 6 x + 16 ?

( )

1

1

10.

11.

2

?2

( )

x 2 ? 6 x ? 16 = 0 ? ( x ? 8)( x + 2 ) = 0 ? x = 8 or x = ?2

(

?x 2?

1

2

= 1 ? log 3 ?? 2 3 ?? = 1 ? x 2 ? 3 = 3 ?

2 ) log 3 x ? ( 3 ) log 3 x = 1 ? log 3 x ? log 3 x

?x ?

1

1

1

2

1

x ? 6 = 3 ? x = 3?6 =

1

5x

4x

3

1

729

log 54

¡Ö 2.0499

13. log 10 x = 17 ? x = 1017

log 7

12. 7 x = 54 ? x = log 7 54 ? x =

14. 5 x = 9 ? 4 x ?

2

2

= 9 ? ( 54 ) = 9 ? x = log 5 9 ? x ¡Ö 9.8467

x

4

15. 10 = e ? x = log 10 e ? x = log e ¡Ö 0.4343

x

16. e ? x = 1.7 ? ? x = ln 1.7 ? x = ? ln 1.7 ¡Ö ?0.5306

1.013

17. ln (ln x ) = 1.013 ? ln x = e1.013 ? x = e e ¡Ö 15.7030

18. 8 x = 9 x ? 1 = ( 98 ) ? x = log 9 1 ? x = 0

x

8

19. 10

x +1

= e ? x + 1 = log e ? x = log e 4 ? 1 = log e 4 ? log 10 ? x = log

4

4

20. log x 10 = ?1.54 ? x ?1.54 = 10 ? x = 10 ? 1.54 ¡Ö 0.2242

1

( ) ¡Ö 0.7372

e4

10

=

1

64

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