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logarithm properties cheat sheet, Cheat Sheet of Mathematics

Great and complete logarithm functions properties cheat sheet

Typology: Cheat Sheet

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Uploaded on 09/02/2019

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PROPERTIES OF LOGARITHMIC FUNCTIONS
EXPONENTIAL FUNCTIONS
An exponential function is a function of the form
(
)
x
bxf =, where b > 0 and x is any real
number. (Note that
(
)
2
xxf = is NOT an exponential function.)
LOGARITHMIC FUNCTIONS
yx
b=log means that y
bx = where 1,0,0
>
>
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 2
10100 =.
x
10
log is often written as just xlog , and is called the COMMON logarithm.
x
e
log is often written as xln , and is called the NATURAL logarithm (note: ...597182818284.2
e).
PROPERTIES OF LOGARITHMS EXAMPLES
1. NMMN bbb logloglog += 2100log2log50log
=
=
+
Think: Multiply two numbers with the same base, add the exponents.
2. NM
N
M
bbb logloglog = 18log
7
56
log7log56log 8888 ==
=
Think: Divide two numbers with the same base, subtract the exponents.
3. MPM b
P
bloglog = 623100log3100log 3===
Think: Raise an exponential expression to a power and multiply the exponents together.
xbx
b=log 01log =
b (in exponential form, 1
0=b) 01ln
=
1log =b
b 110log10 = 1ln
=
e
xbx
b=log x
x=10log10 xe x=ln
xb x
b=
log Notice that we could substitute xy b
log= into the expression on the left
to form y
b. Simply re-write the equation xy b
log= in exponential form
as y
bx =. Therefore, xbb y
x
b==
log . Ex: 26
26ln =e
CHANGE OF BASE FORMULA
b
N
N
a
a
blog
log
log =, for any positive base a. 6476854.0
079181.1
698970.0
12log
5log
5log12 =
This means you can use a regular scientific calculator to evaluate logs for any base.
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PROPERTIES OF LOGARITHMIC FUNCTIONS

EXPONENTIAL FUNCTIONS

An exponential function is a function of the form ( )

x f x = b , where b > 0 and x is any real

number. (Note that ( )

2 f x = x is NOT an exponential function.)

LOGARITHMIC FUNCTIONS

log (^) b x = y means that

y x = b 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: log 10 100 = 2 , since

2 100 = 10.

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 ≈ 2. 718281828459 ...).

PROPERTIES OF LOGARITHMS EXAMPLES

  1. log (^) b MN = log bM +log bN log 50 +log 2 =log 100 = 2

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

2. M N

N

M

log (^) b = log b −log b log 8 1 7

log 8 56 log 87 log 8 = 8 = 

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

  1. M P bM

P log (^) b = log log 100 3 log 100 3 2 6

3 = ⋅ = ⋅ =

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

b x

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

0 b = ) ln 1 = 0

log (^) b b = 1 log 10 10 = 1 ln e = 1

b x

x log (^) b = x

x log 10 10 = e x

x ln =

b x b x =

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

to form

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

as

y x = b. Therefore, b b x b x y = =

log

. Ex: 26

ln 26 e =

CHANGE OF BASE FORMULA

b

N

N

a

a b log

log log = , for any positive base a. 0. 6476854

  1. 079181

log 12

log 5 log 12 5 = ≈ ≈

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. log ( 10 ) 1

2 9 x − =

  1. log 3 15

2 1 3 =

x +

  1. log (^) x 8 = 3
  2. log 5 x = 2

5. log ( 7 7 ) 0

2 5 x −^ x + =

  1. log 3 27 = 4. 5

x

log (^) x 8 =−

8. log 6 x + log 6 ( x − 1 ) = 1

9. log log ( ) 3

1 2 2

2 1

  • = x x

10. log log 2 ( 3 8 ) 1

2 2 x −^ x + =

11. ( ) log ( ) log 1

2 (^33)

1 (^23)

(^1) xx =

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

  1. 7 = 54

x

  1. log 10 x = 17

x x 5 = 9 ⋅ 4

  1. e

x 10 =

  1. = 1. 7

x e

17. ln( ln x ) = 1. 013

x x 8 = 9

1 4 10 e

x

  1. log (^) x 10 =− 1. 54

Solutions to the Practice Problems on Logarithms:

1. log ( 10 ) 1 9 10 19 19

2 1 2 2 9 x −^ = ⇒ = x − ⇒ x = ⇒ x

  1. log 3 15 3 3 2 1 15 2 14 7

2 1 15 2 1 3 =^ ⇒ = ⇒ + = ⇒ = ⇒ =

x x x

x x

  1. log 8 3 8 2

3 x =^ ⇒ x = ⇒ x = 4.^ log^2525

2 5 x =^ ⇒ = xx =

5. log ( 7 7 ) 0 5 7 7 0 7 6 0 ( 6 )( 1 ) 6 or 1

2 0 2 2 5 x −^ x + = ⇒ = xx + ⇒ = xx + ⇒ = xx − ⇒ x = x =

6. log 27 4. 5 log ( 3 ) 4. 5 log 3 4. 5 3 4. 5 1. 5

3 3

3 3 =^ ⇒ 3 = ⇒ = ⇒ x = ⇒ x =

x x x

4

1 2

3 3 2 2 3 log 8 = − ⇒ = 8 ⇒ = 8 ⇒ =

− − x x x x

thenewequation. 3 istheonlysolution totheoriginal equation.

3 2 0 3 or 2 .Note: 2 isanextraneoussolution,whichsolvesonly

log log 1 1 log 1 6 6 0

2 2 2 6 6 6

x

x x x x x

x x x x x x x x

64

1 3 3 2 2 2 3 log 2 3 log 2 3 2 2

log log

(^1212) 2 1 (^12) = ⇒ = ⇒ = ⇒ = = 

− − − x x x x

x

x

x

6 16 0 ( 8 )( 2 ) 0 8 or 2

log log 3 8 1 log 1 2 6 16

2

2 (^223838)

2 2

2 2

x x x x x x

x x x x x

x x

x

729

(^61)

3 3 3

2 (^33)

1 (^23)

1

log log 1 log log 1 log 1 3

6 1

3 2 2 1

3 2

(^12) 3 2 2 1

− −

x x

x x

x x x x x

log 7

log 54 7 = 54 ⇒ x =log 7 54 ⇒ x = ≈

x

17 log 10 x = 17 ⇒ x = 10

14. 5 9 4 9 ( ) 9 log 9 9. 8467

4 4 5

5 4

5 = ⋅ ⇒ = ⇒ = ⇒ x = ⇒ x

x x^ x x

x

  1. 10 = ex =log 10 ex =log e ≈ 0. 4343

x

  1. = 1. 7 ⇒− =ln 1. 7 ⇒ =−ln 1. 7 ≈− 0. 5306

e x x

x

17. ln( ln ) 1. 013 ln 15. 7030

  1. 013 1.^013 = ⇒ = ⇒ = ≈

e x x e x e

18. 8 9 1 ( ) log 1 0

8 8 9

9 = ⇒ = ⇒ x = ⇒ x =

x x^ x

19. 10 1 log log 1 log log 10 log( ) 0. 7372

10

1 4 4 4 4 4 = ⇒ + = ⇒ = − = − ⇒ = ≈

x + (^) e e x e x e e x

  1. log 10 1. 54 10 10 0. 2242
    1. 54
  2. 54 1 = − ⇒ = ⇒ = ≈

− − x x x