Logarithms and Exponentials

Logarithms

100

What is the expanded version of log(ab)?

log(a)+log(b)

200

Write the given exponential form in logarithmic form:

2³=8 

log₂8=3

300

Given log₃81=x; find the value for x

x=4

400

Rewrite the following expression in its completely expanded form:

log₅((3a²)(5/2)(b³))

log₅3+2log₅a+log₅5-log₅2+3log₅b

Exponentials

100

What is the following equivalent to?:

a^m(aⁿ)

 

a^m+n

200

Write log_x125=5 in exponential form. What is the value of x?

5^x=125

x=3

300

What is the value of 5⁰?

Why?

=1

5^1-1=5/5=1

400

Solve for x

e^2x=5

x=ln(5)/2

Solving exponentials and logarithms

100

Solve for x

3log₂(x+3)=12

x=13

200

Solve for z (round to nearest thousandth)

4^z+7+3=80

z=-3.867

300

Find the inverse

y=2^x

ln(x)/ln(2)

400

Solve for x

log₂(4x+1)=3

x=7/4

Real World Applications

100

A population can be modeled as 

P(t)=200(2^t/4)

What is P(12)?

P(12)=1600

200

If a population of bacteria doubles every 3 hours, and initially there are 500 bacteria, write an exponential function to model the population after t hours.

P(t)=500(2^t/3)

300

A car’s value depreciates by 10% per year. If the car’s initial value is $20,000, write an exponential function to model the car's value after t years.

f(t)=20000(0.9)^t

400

The population of a city grows exponentially at a rate of 2% per year. If the current population is 500,000, what will the population be in 15 years?

(Use the formula: f(t)=P₀(1+r)^t

f(15)=905,700