Exponential Funtion
f(x) = b^x
Where b is a real number, b>0, but b does not equal 1.
x can be any real number.
The graph of an exponential function f with base b
It approaches, but does not touch the x-axis. This axis, whose equation is y=0, is a horizontal asymptote.
Transcendental function
Beyond the traditional polynomials and rational functions of algebra.
Is this an exponential function? If so, what is the base?
1.) 2^x
Yes, base is 2
2.) x^2
No
3.) 10^x
Yes, base is 10
4.) 1^x
No (b cannot equal 1)
5.) 3^x+1
Yes, base is 3
6.) (-2)^x
No
7.) -3^x
Yes, base is 3 (-1*3^x)
8.) ?^-x
Yes, base is pi
Natural Base
An irrational number denoted by e
Defined as: e = (1+1/n)^n
Compound Interest- compounding periodically
Formula:
A= P(1+r/n)^nt
A: Accumulated amount
P: Principal
r: interest rate as a decimal
t: time in years
n: the number of compounding periods per year
n times per year and the word that goes with it
12 - monthly
4 - quarterly
2 - semiannually
1 - annually
365 - daily
Compound interest- compounding continuously
Formula:
A= Pe^rt
A: Accumulated amount
P: Principal
r: interest rate as a decimal
t: time in years
Definition of a logarithmic function
For x>0, b>0
b does not equal 1
base is b
Important bases
10" common log
"e" natural log
Domain and range of a log function
Domain: 0 to infinity
Range: negative infinity to positive infinity
When changing from log to exponential or vice versa
Ask what number is the base raised to in order to equal the argument?
Properties for when base is b
1.) logbb^1 = 1
b^1= b
2.) logb^1= 0
b^0 = 1
3.) logbb^x = x
4.) b^logbx = x
Properties for when base is 10
1.) log10b^1 = 1
10^1= 10
2.) log10^1= 0
10^0 = 1
3.) log10b^x = x
4.) 10^logbx = x
Properties for when base is e
1.) logeb^1 = 1
e^1= e
2.) loge^1= 0
e^0 = 1
3.) logeb^x = x
4.) e^logbx = x
Properties for logb
1.) logbb = 1
2.) logb^1 = 0
3.) logbb^x = x
4.) b^logbx = x
Properties for log (log10)
1.) log10 = 1
2.) log1 = 0
3.) log10^x = x
4.) 10^logx = x
Properties for ln
1.) lne = 1
2.) ln1 = 0
3.) lne^x = x
4.) e^lnx = x
Product rule
logbMN = logbM + logbN
Quotient rule
logbM/N = logbM - logbN
Power rule
logbM^P = PlogbM
Change of base formula
logba = loga/logb or lna/lnb
Method one to solving exponential equations
Use like bases when you can write both sides of the equation with the same base
for example:
5^3x-6 = 125
5^3x-6 = 5^3
3x-6 = 3
3x = 9
x=3
Method two to solving exponential equations
Use logarithms when you can't use like bases
for example:
1.)
10^x = 8000
log(10^x) = log(8000)
x = log(8000)
2.)
e^2x = 8
lne^2x = ln8
2x/2 = ln8/2
x= ln8/2
Solving log equations
1.) for logbM = C
c is a number
use the log definition to solve : y=logbx <--> b^y=x
2.) for logbM= logbM= logbN
then M=N
where M,N>0
Population growth
P= P0e^kt
P: population at time, k
P0: initial population
K: growth rate
K>0
Population decay
P= P0e^kt
P: population at time, k
P0: initial population
K: decayrate
K<0