/5.5__Properties_and_Laws_of_Logarithms

5.5 – Properties and Laws of Logarithms RECALL OUR THEME: A logarithm is an exponent!!

Properties of Logarithms: 2. loga a  1 3. loga a x  x

1. loga 1  0

4. aloga x  x

EXAMPLE 1: Evaluate the following: a. log3 37

c. log15 1

1 e. log 4   2

g. 10log50

b. log4 16

d. log9 3

f. log49 7

h. 5log5 

1. ln 1  0

Properties of Natural Logarithms: 2. ln e  1 3. ln e x  x

EXAMPLE 2: Evaluate the following:  1  a. lne8 b. ln  2  e 

c. eln 6

EXAMPLE 3: a. Express the equation in exponential form and solve: log  x  3  1

b. Express the equation in exponential form and solve: ln x  1  4

c. Express the equation in logarithmic form and solve: e x 1  0.5

4. eln x  x

d. ln5

♥ Since “Logarithms are Exponents” the Laws of Exponents give rise to the Laws of Logarithms.

Laws of Logarithms Let a be a positive number, with a  1 .

Let A, B, and C be any real numbers with A  0 and B  0 . loga  AB  loga A  loga B OR ln AB  ln A  lnB  A loga    loga A  loga B B

 

loga AC  C  loga A 

OR

OR

 A ln   ln A  lnB B

 

ln AC  C  ln A

We will eventually be solving logarithmic equations, and we will need to know how to expand or combine logarithmic expressions.

EXAMPLE 4: Evaluate each expression: a. log3 3  log3 9

b. log4 32  log4 2

EXAMPLE 5: Use Laws of Logarithms to expand each expression: x 10 a. log 5 b. log2  xy  2

EXAMPLE 6: Combine into a single logarithm (simplify): a. log8 x  3log x  log2x2

b. 2 ln x  2ln y  3ln z 

c.

c. log6

2 log8 3

x 1 x 1