Study Flashcards
The function f(x) = a^x with a > 0 and a ≠ 1 is
The exponential function with base a
The exponential function with base a
f(x) = a^x with a > 0 and a ≠ 1
What is x^2 and what is 2^x
Power function and exponential function
Power function and exponential function
x^2 and 2^x
Domain and range of exponential functions
(-∞, ∞) and (0, ∞)
(-∞, ∞) and (0, ∞)
Domain and range of exponential functions
How do you graph y=a^x?
Y intercept at 0,1
Point at (1,a)
Lim a^x = ∞
x —> ∞
Lim a^x = 0
x —> -∞
How do you graph the opposite of y = a^x?
Y intercept (0,1)
Point at (1,a)
Point at (-1, 1/a)
Lim a^x = 0
x—> ∞
Lim a^x = ∞
x—> -∞
Y intercept at 0,1
Point at (1,a)
Lim a^x = ∞
x —> ∞
Lim a^x = 0
x —> -∞
y = a^x
Y intercept (0,1)
Point at (1,a)
Point at (-1, 1/a)
Lim a^x = 0
x—> ∞
Lim a^x = ∞
x—> -∞
Opposite of y=a^x
What shift is this?
y=a^x-c
Shift right c units
How do you shift right c units in an exponential function?
y=a^x-c
Reflections in exponential functions
y=-a^x (x-axis)
y=a^-x (y-axis)
What is this transformation?
y=Ca^x
Stretch if C>1
Shrink if 0
Examples of exponential growth / decay
Capital, population, radioactive isotopes
Capital, population, radioactive isotopes
Examples of exponential growth / decay
Discrete compound interest formula
A=P(1+r/n)^nt
A=P(1+r/n)^nt
Discrete compound interest formula
What is r and n in discrete compound interest formula?
N is number of times compounded like (annually, semi-annually, quarterly, monthly, or daily) and r is perfentage
Continuously compounded interest formula
A=Pe^rt
Lim (1+1/h)^h = e
A=Pe^rt
Lim (1+1/h)^h = e
Continuously compounded interest formula
Exponential growth and decay formula
A(t)=A(o)e^kt or A(o)(e^k)^t
Initial amount is A(o)
K is relative time rate of graph (k > 0) for growth and (k < 0) for decay
A(t)=A(o)e^kt or A(o)(e^k)^t
Initial amount is A(o)
K is relative time rate of graph (k > 0) for growth and (k < 0) for decay
Exponential growth and decay formula
How do you graph:
y=logaX
X intercept (1,0)
Domain: (0, ∞)
Range: (-∞, ∞)
X intercept (1,0)
Domain: (0, ∞)
Range: (-∞, ∞)
y=logaX
The logarithm with base a is defined by
y=logaX iff x=a^y
y=logaX iff x=a^y
Logarithm with base a
Convert to logarithmic form:
4^3=64
(1/2)^4=1/16
a^-2=7
3=log4(64)
4=log1/2(1/16)
-2=loga(7)
3=log4(64)
4=log1/2(1/16)
-2=loga(7
4^3=64
(1/2)^4=1/16
a^-2=7
Convert to exponential form:
log3(243)=5
log2(5)=x
loga(N)=x
243=3^5
5=2^x
N=a^x
243=3^5
5=2^x
N=a^x
log3(243)=5
log2(5)=x
loga(N)=x
The number loga(x) is
The power that we raise a to in order to get x
The power that we raise a to in order to get x
loga(x)
log5(25)
log2(16)
log1/3(9)
log7(7)
log16(1)
log4(1/2)
log5(5^2)=2
log2(2^4)=4
log1/3(3^2)=log1/3^-2
log7(7)=1
log16(1)=0
log4(1/2)=-1/2
log5(5^2)=2
log2(2^4)=4
log1/3(3^2)=log1/3^-2
log7(7)=1
log16(1)=0
log4(1/2)=-1/2
log5(25)
log2(16)
log1/3(9)
log7(7)
log16(1)
log4(1/2)
How is log1/3(9) evaluated?
- ) log1/3(3^2) is written
- ) log1/3((1/3)^-1)^2 is written since 3 = 1/3^-1
- ) answer is log1/3(1/3^-2)
Basic properties of logarithms
loga(a)=1
loga(1)=0
loga(a^x)=x for any real x
a^loga(x)= x for any x > 0
loga(a)=1
loga(1)=0
loga(a^x)=x for any real x
a^loga(x)= x for any x > 0
Basic properties of logarithms
How do you find the domain of a logarithmic function?
The numbers inside the parenthesis must be set to 0
What is the domain of:
g(x) = logz(x+1/x-3)
(-∞,-1) U (3, ∞) since the numerator and denominator have to be positive
What is “the exponential function”
e^x
e^x
“The exponential function”
Common logarithm and natural logarithm difference
Common is log base 10 “logx”
Natural is log base e “lnx”
Evaluate:
lne^4
4
Logarithm rules
loga(xy)
loga(x/y)
loga(x^y)
loga(x) + loga(y)
logaX - loga(y)
y • loga(x)
loga(x) + loga(y)
logaX - loga(y)
y • loga(x)
loga(xy)
loga(x/y)
loga(x^y)
Exponent rules
a^x • a^y
ax/ay
(a^x)^y
a^x+y
a^x-y
a^xy
a^x+y
a^x-y
a^xy
a^x • a^y
ax/ay
(a^x)^
Change if base formula
loga(x) = ln(x)/ln(a)
loga(x) = ln(x)/ln(a)
Change of base formula
Radioactive decay formula
h=-ln2/k
h=-ln2/k
Radioactive decay formula
