Exponential Functions
If $5400 is invested at an interest rate of 3.5% per year, compounded continuously, find the value of the investment after 2 years.
$5791.54
Write the equations in exponential form
(a) ln (5) = 3y
(b) ln(t+1) = -1.
(a) 5=e^{3y}
(b) t=e^{-1} -1
Use laws of logarithms to evaluate the expression
log_3(100) - log_3(18) - log_3(50)
=-2
Solve:
e^{1-2x} = e^{4x-7}
x=4/3
Does the function represent growth or decay?
f(x)=16(8/3)^x
Exponential growth
If $750 is invested at an interest rate of 3.75% per year, compounded quarterly, find the value of the investment after 5 years.
$903.88
Write the equation in logarithmic form
(a) 3^{2x}=10
(b) 10^{-4x}=0.1
(a) x=1/2*log_3(10)
(b) x=1/4
Use the Laws of Logarithms to expand the expression
log(AB^2)
log(A) + 2*log (B)
Find the exact solution of the exponential equation in terms of logarithms.
2(5+3^{x+1})=100
x=log_3(45) - 1
The population of the city of Martin was approximately 12,420 in the year 2005 and has been continuously growing at a rate of 1.6% each year.
What was the population in 1996?
approximately 10,755 people
What amount was deposited if the current value is $100,000 and interest is paid at a rate of 8% per year, compounded monthly, for 5 years?
$67,121.04
Find the domain of the following function
f(x)=log_5(8-2x)
(-oo,4)
Simplify the following expression
e^{2x}*e^{3y}*e^{-7z}
e^{2x+3y}/e^{7z}
Solve the equation.
log(x)+log(x+21)=2
x=4
Does the function g(x)=24e^(-3x) represent exponential growth or decay?
Exponential decay
If $925 is invested at an interest rate of 2.5% per year, find the amount of the investment at the end of 10 years, given that it is being compounded semiannually.
$1185.88
Find the domain of the function
f(x)=ln(x-3)-1
(3, oo)
Use the Laws of Logarithms to condense the expression.
3 ln 2 + 2 ln x - 1/2(ln(x+4))
ln((8x^2)/(x+4)^(1/2))
Solve the equation
log_2(x-9)+log_2(x+3)=log_2(13)
x=10
The half-life of strontium-90 is 28 years. How long will it take a 50-mg sample to decay to a mass of 32 mg?
approximately 18 years
A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function m(t)=13e^{-0.015t} where m(t) is measured in kilograms.
(a) Find the mass at time t = 0 .
(b) How much of the mass remains after 45 days?
(a) 13 kg
(b) 6.62 kg
For the given function, state the following:
> domain
> range
> equation of the asymptote
> x-intercept
> y-intercept
f(x)=log_2(x-3)+4
Domain: (3, oo).
Range: (-oo, oo).
Equation of the asymptote: x=3.
x-intercept: (3.0625, 0).
y-intercept: "none".
Use the Change of Base Formula to show that
log e = 1/ln(10)
log e = ln(e)/ln(10) = 1/ln(10)
Solve for x .
ln(x-1/2)+ln(2)=2ln(x)
x=1
You plant a sunflower seedling in a community garden. The height (in centimeters) of the sunflower after t weeks can be modeled by the logistic function h(t)=(256)/(1+13e^(-0.65t)).
How long does it take for the sunflower to reach a height of 200 centimeters?
approximately 5.9 weeks