Transformations
Name the transformation(s).
f(x)=2^x
g(x)=2^(x+3)
Horizontal shift left 3
Evaluate.
log_5(1)
0
Solve:
x=log_5(125)
x=3
Solve. Round to 3 decimal places.
e^(3x)=15
x=0.903
What 3 things do you look for to know you have fully EXPANDED a logarithm?
1) Addition and subtraction signs
2) Coefficients
3) Multiple logs/ln
Tell whether the population is an exponential growth or decay function.
P(t)=102,324(1.025)^t
Growth function
Name the transformation(s).
f(x)=log(x)
g(x)=-log(x)+4
Reflection across the x-axis
Vertical shift up 4
Evaluate.
lne^(-7)
-7
Solve.
log_x(1/64)=-2
x=8
Solve. Round to 3 decimal places.
4(5^x)=32
x=1.292
Condense the logarithm.
2log(5x)+3log(x)-log(y)
log((25x^5)/(y))
The given function describes the population of elephants once they were moved to a safe habitat after t years.
Find how many elephants will be in the habitat after 3 years? Round to the nearest whole number.
f(t)=(1500)/(1+38e^(-0.9t)
About 422 elephants
Given the transformations and parent function, create an equation.
Parent function:
f(x)=5^x
1)Horizontal shift right 2
2) Vertical compression by 1/3
3)Reflection across the y-axis
g(x)=(1/3)5^(-x-2)
Evaluate.
log_64(8)
1/2
Solve. Round to 3 decimal places.
7-4logx=10
x=0.178
Solve.
3^(2x-1)=27
x=2
Expand.
ln(sqrt((x^3)/y))
1/2(3lnx-lny)
The population of Pikeville is 425,000 and it is increasing at a rate of 4.2% each year. Predict the number of years until the population will be 750,000. Round to two decimal places and include units.
P(t)=P_0(1+r)^t
13.8 years
Name the transformation(s).
f(x)=e^x
g(x)=4e^(2x)-3
Vertical stretch by 4
Horizontal shrink by 1/2
Vertical shift down 3
Evaluate.
log(1/(sqrt(10,000)))
-2
Solve. Round to 3 decimal places.
4log(x-1)=2
x=4.162
Solve.
24(1/2)^(x/3)=12
x=3
Expand the logarithm.
log((100x)/(z^2y^4))
2+logx-2logz-4logy
A single-cell amoeba doubles every 3 days. How long would it take one amoeba to produce a population of about 10,000 amoebae?
Round to one decimal place.
39.9 days
Given the transformations and parent function, create an equation.
Parent function:
f(x)=ln(x)
1.) Reflect across x-axis
2.) Vertical shift up 3
3.) Horizontal shift left 5
4.) Vertical compression by 1/4
g(x)=-1/4ln(x+5)+3
Evaluate.
log_6(1/36^(1/5))
-2/5
Solve.
log(x-4)+log(x+5)=1
x=5
Solve. Round to 3 decimal places.
(4-2.375/40)^(8t)=28
t=0.304
Condense.
1/2[3log(x+1)-logx-log(x-3)]
log(sqrt(((x+1)^3)/(x^2-3x)))
Suppose the half-life of a certain radioactive substance is 24 days and there are 10 grams present initially. Find the time when there will be 2 grams of the substance remaining.
Round to one decimal place.
55.8 days