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