Log Properties
Find the value of
log_3(1/81)
-4
The wingspan w(x) of a bird in feet can be predicted based on the bird's weight in pounds. Use the equation below to predict the wingspan of a bird that weighs 2 pounds.
w(x)=1.5+2ln(x)
w(2)=1.5+2ln(2)
w(2)=2.9 feet
You solve log_6x+log_6(x+9)=2 and get the solutions x=-12 and x=3 . Which of these is a real solution?
x=3
Write in logarithmic form:
3.5=(1.061)^t
log_(1.061)(3.5)=t
What is the domain and range of y=log_7x ?
Domain:
(0, \infty)
Range:
(-\infty, \infty)
Rewrite in logarithmic form and solve to the nearest tenth:
e^(2t)=16
ln(16)/2=(2t)/2
t=1.4
The reference intensity I_0=1xx10^-12 watts per square meter measures a sound that is just barely audible and has an intensity level of 0 decibels. The intensity level β, measured in decibels (dB), is defined as: d\beta=10log(I/I_0) .
The average snore has a sound intensity of 3.16xx10^-6 W/m^2 . What would be the intensity level in decibels? Round to the nearest whole number.
d\beta=10log((3.16xx10^(-6))/(1xx10^(-12)))
d\beta=65
Condense into one log:
log_4x+log_4(x-12)=3
log_4(x)(x-12)=3
log_4(x^2-12x)=3
How long will it take an account with $100 and an 8% annual growth rate to grow to $150?
150/100=(100(1+.08)^t)/100
1.5=(1.08)^t
log_(1.08)(1.5)=t
t=5.3 years
Solve for x:
log_18x+log_18(x-3)=1
log_8(x^2-3x)=1
18^1=x^2-3x
0=x^2-3x-18
0=(x-6)(x+3)
x=6, x=-3
Check: x=6 only
Expand:
log(x^2/w^5)
2logx-5logw
SET UP THE EQUATION ONLY:
The temperature F(t) of an object can be modeled by the function below where F_S is the surrounding temperature, F_0 is the initial temperature of the object, and k is the cooling constant. When a steak comes out of the oven it is 140^oF. The steak sits to cool in a room at a temperature of 71^oF. If the cooling constant is k=0.082, how long will it take for the steak to cool to 100^oF to the nearest minute?
F(t)=F_S+(F_0-F_S)e^(-kt)
100=71+(140-71)e^(-0.082t)
Convert to exponential form:
log_20(x^2-x)=1
20^1=x^2-x
An account with a 8.6% growth rate is compounded continually. How long will it take the account to double?
A_0=100, A(t)=200
200/100=(100e^(0.086t))/100
2=e^(0.086t)
ln(2)/0.086=(0.086t)/0.086
t=8.1 yrs
Write in exponential form:
log_((1+.06/4))(2)=4t
2=(1+.06/4)^(4t)
Condense:
3(logp+logr)
log(p^3r^3)
or
log[(pr)^3]
SOLVE:
The temperature F(t) of an object can be modeled by the function below where F_S is the surrounding temperature, F_0 is the initial temperature of the object, and k is the cooling constant. Molten glass has a temperature of 2900^oF. A glass worker is working with some molten glass in a room that is 85^oF. The cooling constant is k=0.036. If the glass will harden again at 2600^oF, how many minutes does the the glassworker have to work with the glass?
F(t)=F_S+(F_0-F_S)e^(-kt)
2600=85+(2900-85)e^(-.036t)
2515/2815=((2815)e^(-.036t))/2815
0.89=e^(-.036t)
ln(0.89)/-.036=(-.036t)/-.036
t=3 min
Solve for x:
log_2x+log_2(x+4)=5
log_2(x^2+4x)=5
2^5=x^2+4x
32=x^2+4x
0=x^2+4x-32
0=(x+8)(x-4)
x=-8, x=4
Check: solution is x=4 only
An account is started with $3,000 and it has a 2.25% growth rate that is compounded monthly. How long will it take the account to reach $4,000?
4000/3000=(3000(1+.0225/12)^(12t))/3000
1.3=(1+.0225/12)^(12t)
log_(1+.0225/12)(1.3)/12=(12t)/12
t=12.8 yrs
DOUBLE JEOPARDY
The cost of a piece of art is $2000 in 2022 and it appreciates at 4% per year.
In what year will the art be worth $2500?
2500/2000=(2000(1.04)^t)/2000
1.25=(1.04)^t
log_(1.04)(1.25)=t
t=5.7 years
2022+6=2028