Unit 4: Exponential Functions
Vertical Asymptote of Logarithmic Function
x=0
Horizontal Asymptote of Exponent Function
y=0
Logarithm Transformation Formula
y=alogb(x-h)+k
Inverse of y=a*b^x
y=logbx where x>0
Inverse Function
A function in which two variables are inversely proportional.
Exponential Function
A function of the form f(x) = ab^x, where a and b are real numbers with a ≠ 0, b>0, and b ≠ 1.
Logarithmic Function
A function of the form f(x)=logbX, where b≠1 and b>0, which is the inverse of the exponential function f(x)=b^X.
Horizontal Asymptote
A horizontal line which a graph approaches.
r
Constant percentage rate per year.
Natural Logarithms
ln x=logex➡️(base e)➡️ln and e are inverse, so they will cancel each other.
Exponential Population Model
p(t)=P0(1+r)^t
Formula for Exponential Function
f(x)=a*b^x
Key Points of Exponential Function
(0,1) (1,b)
Key Points of Logarithmic Function
(1,0) (b,1)
How to Solve a Logarithmic Equation (With multiple logs and ln)
*One log (ln) on each side of = sign will cancel out.* 1. Rewrite with one log: Remember you must condense the logarithmic expression. 2. Once one log (ln) on each side of the = sign, cancel out the log (ln) 3. Simplify and solve.
Exponential Growth
-Growth pattern in which the individuals in a population reproduce at a constant rate. -b>1.
h
-How many units the function is shifted left or right. -Opposite.
k
-How many units the function is shifted up or down -Same.
a
-Initial Value. -y intercept, where x=0.
(1+r)
-The base. -The b value.
t
-The exponent. -The x value. -A measure of time in years.
P0
-The initial amount/population. -The a value.
Exponential Decay
-When an initial amount decreases by the same percent over a given period of time. -0<b<1.
Steps to Evaluate Logs
1. Change log to exponential form. 2. If bases are the SAME; then they will cancel out. 3. Solve for x.
How to Solve a Logarithmic Equation
1. Isolate b^x (get b^x by itself) 2. Use exponential property: Same bases will cancel. 3. Solve for x. *Only 1 log can change to exponential form.*
How to Determine if a Function is Growing or Decaying
1. Remember y=a*b^x. 2. Look at the base, where a>0. If the base is greater than 1, then it is a growth. (b>1) If the base is less than 1, then it is a decay. (0<b<1) If the a value is negative, then the function is neither. This is because a negative a value reflects the function across the x-axis. This shows no decay or growth because a<0. 3. In order to make negative exponent into positive ones, multiply them by the reciporcal.
How to Solve a Logarithmic Equation (If bases are not the same)
1. Use log or ln to solve for x. 2. Isolate b^x (get b^x by itself) 3. Introduce log to BOTH sides of the = sign. 4. Use log properties to bring x to the front. (Power rule) 4. Solve for x. *In and e are inverses! They will cancel.*
Basic Properties of Log
1. logb1=0, b^1=0. 2. logbb=1, b^1=b 3. logbbx=x, b^x=b^x, the bases cancel. x=x. 4. blogbx=x. The bases cancel, x=x.
Vertical Asymptote
A vertical line that a graph approaches but never crosses.
Common Logarithms
Base=10. y=logx= y=log10x
Domain and Range of Exponential Parent Function
Domain: (-∞,∞) Range: [0,∞)
Domain and Range of Logarithmic Parent Function
Domain: (0,∞) Range: (-∞,∞)
a (Transformation)
How much the function is vertically strectched/shrunk.
Inverse of Exponential Function
Logarithmic Function
Power Rule
Move the exponent to the front of the log. logbR^p➡️plogbR
Product Rule
Multiplying powers with the same base, keep the base the same, add the exponents. logb(RS)➡️logbR+lobgbS
-a
Reflection across the x-axis.
-x
Reflection across the y-axis.
Doubling and Half-Life Model
f(x)=P0(1+r)^x/c f(x)=Final amount/population. P0=Initial amount/population. (1+r)=Base. r=Constant percentage rate each year. x=Exponent. c=What the exponent is divided by; the amount of time.
Transformation Formula of Exponential Function
f(x)=a(b)^x-h +k
If the base is not 10....
The Change of Base Formula must be used.
b (Transformation)
The base.
b
The base. (Always positive.)
x (Transformation)
The exponent.
x
The exponent. (Must be a variable.)
p(t)
The final amount/population.
Natural Base
The irrational number e, which is approximately equal to 2.718281828...
Domain
The set of x-coordinates in a relation.
Range
The set of y-coordinates in a relation.
Quotient Rule
To divide when two bases are the same, keep the base and SUBTRACT the exponents. logb(R/S)➡️logbR-logbS
Change of Base Formula
b is on bottom-or-b goes in the basement.
y=logbx
b^y=x
Half Life Model
f(t)=(1+r)=1/2 b=1/2
Doubling Model
f(t)=(1+r)=2 b=2