Logs Logs Logs

Log Properties

100

2log3 - log8

log (9/8)

200

3ln2 + 5ln2 - ln3

ln (256/3)

300

1/2log(4) + log(2xy)

log(4xy)

400

2log₅(5x) + log₅y - log₅(x+1)

log₅(25x²y/x+1)

500

5(log2x - logy) - (log3 + 2log5)

log(32x⁵/75y⁵)

Solving Case 1

100

log₅(5x+7) - log₅(8) = log₅(49)

x=77

200

log₅(x-8) - log₅(2) = log₅(78)

x=164

300

log(2+x) - log(x-2) = log(3)

x=4

400

log₂(x+8) - log₂(3-x) = log₂(10)

x=2

500

log₄(x+1) - log₄(7) = log₄(2x-3)-log₄(3)

x=24/11

Solving Case 2

100

log₄(-4x) - log₄(6) = 1

x=-6

200

log₈(5) + log₈(5x-1) = 2

x=69/25

300

log₅(2) + log₅(x+7) = 1

x= -9/2

400

log(x-3) - log(x) = 2

x= -1/33

500

log₃(x+4) - log₃(x-4) = 3

x= 56/13

Solving Case 3

100

4^2x-1 = 64

x=2

200

32^x-2 = 64

x= log₃₂64 + 2

x= 3.2

300

3^x - 12 = 11

x= log₃23

x= 2.85

400

5*2^x +10 = 40

x= log₂6

x= 2.58

500

5 - 4^x *2 = -17

x= log₄11

x= 1.73

Application

100

A town has a population of 20,000 and grows at 4% every year.  How long until the town's population reaches 28,000? 

Round to 2 decimal places.

8.58 years

200

Jenny bought a new car for $19,000.  The value of the car depreciates at 11.25% per year.  In how many years will the car be worth $10,000? 

Round to 2 decimal places.

5.38 years

300

Gaby places $1,400 in a savings account with an annual interest rate of 8.25%.  How long does it take until her investment doubles.

Round to 2 decimal places.

8.74 years

400

An element with a mass of 800 grams has a half life of 5 minutes.  How long until the element decays to 10 grams.

Round to 2 decimal places.

31.61 minutes

500

A bank account with $2,000 grows continuously at a rate of 1.2% per year.  How many years until the bank account has $3,000.

Round to 2 decimal places.

33.79 years