2c1 Properties of Logs
What is equivalent to logbM+logbN?
Product Rule:
logb(MN)
What equation do you get when you divide both sides of 20e2t=100 by 20?
e2t=5
What is the general formula for continuous exponential growth or decay?
A=A0ekt
Explain why the one-to-one property of exponential functions allows us to conclude that if ax=ay, then x=y.
Because exponential functions with the same positive base a≠1 always produce unique outputs for each input, so equal outputs imply equal inputs.
What is the population after 5 years if a town starts with 1,000 people and grows at 8% per year?
p(t)=1000(1.08)t
P(5)=1000(1.08)5=1469.3
What is the expanded form of log(x3y/z2)?
3log x + log y - 2log z
What is the solution to e2t=5?
t=1/2ln(5)
What term describes the amount of time it takes for a quantity to become twice its original amount in an exponential growth model?
Doubling Time
How can you tell whether a situation represents exponential growth or exponential decay just by looking at its equation or context?
In the form y=a(1+r)t, if r>0, the function models growth; if r<0, it models decay. In context, quantities increasing at a constant percentage rate grow exponentially, while those decreasing at a constant percentage rate decay exponentially.
If a car is worth $500 when it’s 0 years old and loses 15% of its value each year, what will it be worth after 4 years?
A=500(0.85)4
261.2~261
What is the condensed form of log3+2logx?
log(3x2)
What is the solution to 3x is equal to square root 3 using the one-to-one property of exponents?
x=1/2
What term describes the time required for a quantity undergoing exponential decay to decrease to half of its original value?
Half-life
Why are the properties of logarithms useful when solving exponential or logarithmic equations?
They allow us to simplify complex logarithmic expressions, separate variables, or combine multiple logs into a single one — making equations easier to solve using algebraic methods or the definition of a logarithm.
The half-life of carbon-14 is 5730 years. If a fossil has 25% of its original carbon-14, how old is it?
k=ln(1/2)/5730=-0.000121
t=ln(0.25)/-0.000121
Answer= 11,460
How can you express 5log2 as a single logarithm?
log(25)=log(32)
What is the solution to 2x−1=23x+2 using the one-to-one property of exponential functions?
x=-3/5
What is the exponential growth model for an initial population of 200 that increases by 5% each year?
Bonus: Solve for three years
p(t)=200(1.05)t
Write out the Logarithm Properties (Product rule, Quotient Rule, Power Rule) and explain them.
PRO: logb(MN)=logbM + logbN The log of a product equals the sum of the logs.
QUT: logb(M/N)= logbM - logb N The log of a quotient equals the difference of the logs.
POW: logb(Mx)= logbM The log of a power equals the exponent times the log.
A radioactive element decays according to N(t)=N0e−0.01386t. How long will it take for half of the element to decay?
Set N/N0=1/2:
1/2=e-0/01386t
t=ln(1/2)/-0.01386=50.106
50 is the answer
What is the expanded form of log ((xy1.5)2/z)
2 log x +3 log y -log z
What is the value of tt that satisfies −2=−3(2t+1) after simplifying?
t=0
What is the decay constant for Carbon-14, which has a half-life of 5730 years?
k=ln(1/2)/5730
A bacteria culture grows continuously at a rate of 5% per year. How long will it take for the population to double?
2=e0.05t- ln(2)=0.05t
t=0.693/0.05
13.86 years