Logarithm Properties With Examples

 

Properties of Logarithms A Logarithm is an Exponent: Exponent: It is the exponent we put on some base to get x. x >0  

Common Logarithm Base 10

log  x =  y   1 0 y   =  x  

l n  x =  y   e y   =  x  

log  x =  y  

log 1 = 0

ln 1 = 0

log 1 = 0

log 10 = 1

ln e = 1

log  b = 1

log 10 x = x

for all x 

log x

x>0

10

General Logarithm Base b

 Natural Logarithm Base e

 = x

ln e x = x

 y

 b

 b

 x

log  b b  = x

for all x 

ln x

e

bb   =  x  

 = x

0

x>

b

for all x 

log x

 = x

x>

0

Product Property log AB = log A + log B and log A + log B= log AB

Product Property ln AB = ln  A + ln B and ln A + ln B = ln AB

Product Property log  bAB = log  bA + log  bB

Quotient Property

Quotient Property

Quotient Property

log

A B

 = log A − log B

ln

A B

and log  bA + log  bB= log  bAB

 = ln A − ln B

log  b

A B

 = log  bA − log  bB

and

and log A − log B= log

A B

 

and

ln A − ln B= ln

Power Property log Bt  = t  log  log B and t  log  log B= log Bt  

A B

 

log A − log B= log  b

 b

A  b

B

 

Power Property t  log B  = t  log  log B

Power Property ln Bt  = t  ln  ln B and t  ln  ln B= ln Bt  

 b

 b

and t  log  log  bB= log  bBt  

Some misconceptions: 1) log (a (a + b) = log a + log log b NOT TRUE 

log (a + b) ≠  log a + log b

log (a − b) ≠ log a − l  log og b But what does “log (a + b)” or “log (a (a − b)” mean? The LOG IS AN EXPONENT. Therefore, log (a + b) must be the exponent we put on 10 to get (a + b). y y i.e. y = log (a + b) means 10  = a + b. Similarly, y = log(a− b) means 10  = a − b 2) log (ab) ≠ (log a)(log b) t

3) log ab  ≠ t log ab t

and

⎛ a ⎞ log a log ⎜   ⎟ ≠    ⎝  b ⎠ log b t

and

1 ⎛ 1 ⎞ log ⎜   ⎟ ≠    ⎝ a ⎠ log a t

Use product property: log ab  = log a + log b  

 but log(ab)  does = t log(ab)

 

Examples: Use properties. Be careful of negative signs signs.. There’s several ways to do these problems, just  be careful using the correct properties.  x

1.  Expand

log

 x

7

 y 5

16 x  z

log

7

 y

3

16 x 5 z 6

 

i.e. Break up or ‘pull apart’ this expression

3

= log( x 7

6

 y

3

) − (log16 x  z )   5

6

by division (quotient) property

multiplication (product) (product) property = log( x 7 ) + log(  y 3 ) − [log(   16) + log( x 5 ) + log( z 6 )]   by multiplication  Notice the parenthesis after the minus sign. = 7 log x +

3

= 2 log x +

3

2

2

log y −  log 16 − 5 log x − 6 log z  

by the power property. Notice log  y 3 = log y

3

2

 

log    y − log16 − 6 log z   combine the log x terms.

2.  Write as a single log. Simplify when possible. In this case we are ‘Putting Together” the expression to get one log

9 log  x −

1 3

log    y − 5 log z + 2  

= log x 9 − log y

1

= log x 9 − (log y

− log z 5 + log100  

3

1 3

+ log z 5 ) + log100  

1

= log x 9 − log( y  x

= log

1

 y

= log

3

3

* z 5 ) + log 100  

+ log 100   * z

100 x 1

3

9

 y  z

5

9

  5

Use power property first Notice that log100 = 2 Now we have all log terms. Use quotient and product properties to combine. I factored out the negative in the Second and third terms first.