Converting to logs
•  Example: 16 = 4x
Converting to exponents
•  Example: 3 = log4x
•  Example: 17 =
3x
•  Example: 4 = log5x
•  Example: 183 = ex
•  Example: lnx = 2
Change of Base Formula
Allows us to rewrite logs in terms of base 10 or e
so we can calculate the value of the log.
change of base formula : log a x =
log x
log a
ln x
for base e : log a x =
ln a
logb x
logb a
for base 10 : log a x =
Properties of Logs
log10=
log10x=
log1=
10logx =
lne=
lnex=
ln1=
elnx =
Example: Find the value log428
Condensing Log Expressions
Example: condense: log5x - log5y
Example: Condense: 3lnx + 4 ln(y+4)
Expanding Log Expressions
•  Use the properties of logs to rewrite one log
expression into multiple log terms.
example: Expand log812y
example : expand ln
x3 y
z6
1
Solving exponential Equations
x
1.
⎛1⎞
5 ⎜ ⎟ = 64
⎝4⎠
2.
6e 2 x + 10 = 46
3.
10e − x + 45 = 83
Solving Log Equations
Example 1: log(4x + 2) – log(x – 1) = 1
Example 2: ln(x – 2) + ln(2x – 3) = 2lnx
Interest
Example
r
•  Compound interest: A = P ⎛⎜1 + ⎞⎟
⎝ k⎠
kt
How long will it take for an investment of $15,000 to grow to
$27,000 if interest is compounded quarterly with a 7.5%
interest rate?
•  Continuously Compounded interest: A=Pert
A: final amount
P: principal
r: interest rate
k: number of payments per year
t: number of years
How long will it take if the interest is compounded
continuously?
Graphs
Transformations
6
6
5
5
4
4
y = 2x
y = 2x+2
3
2
3
2
1
1
−7
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
6
−1
−1
y=
1
2
x
6
6
1
y = 2i
2
5
4
3
x
5
4
3
2
2
1
−7
−6
−5
−4
−3
−2
−1
1
1
−1
2
3
4
5
6
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
−1
2
Transformations
1
−7
y=
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
−1
-2x
−2
−3
−4
−5
−6
x
y=
4
1
−3
2
3
2
1
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
7
−1
−2
−3
−4
3