Pre-Calc: Logarithms

Converting Log and Exponential Form

100

Convert to Logarithmic Form:

4^y=x

log_4(x)=y

200

Convert to Exponential Form:

2log_(3x)(5-z)=4y

(3x)^(2y)=5-z

or

(3x)^(4y)=(5-z)^2

300

Convert to Exponential Form:

log_(x-1)(4)=2y

(x-1)^(2y)=4

400

Convert to Logarithmic Form:

e^(x+4)=2y-7

ln(2y-7)=x+4

500

Convert to Exponential Form:

log_2(8)=x

2^x=8

Solving Log Equations with Properties

100

Solve for x:

log_5(x+6)=log_5(3x)

x=3

200

Solve for x:

3log(4)-log(x)=log(2)

x=32

300

Solve for x:

2ln(2)+ln(x)=ln(x+12)

x=4

400

Solve for x:

2log(x)=log(3x+4)

x=4

-1 is an extraneous solution

500

Solve for x:

log_3(x-3)=log_3(55)

x=58

Expanding/Condensing Logarithms

100

Completely Expand the Logarithm:

log((2x)/y)

log(2)+log(x)-log(y)

200

Condense the Logarithms:

3log(x)-4log(2)-5log(z)+3log(3)

log((27x^3)/(16z^5))

300

Condense the Logarithms:

2ln(3)-2ln(x)+4ln(y)

ln((9y^4)/x^2)

400

Completely Expand the Logarithm:

ln((3x^6y^7)/(7z^5))

ln(3)+6ln(x)+7ln(y)-ln(7)-5ln(z)

500

Condense the Logarithms:

log_3(2x)-log_3(5y)

log_3((2x)/(5y))

Solving Exponential Equations with Logs

100

Solve Using Logarithms:

3^(7x)=108

x=0.61

200

Solve using Logarithms:

e^(7x)=60

x=0.58

300

Solve by Converting:

log_4(x)=5

x=1024

400

Solve using Logarithms:

4^(x-2)=100

x=5.32

500

Solve using Logarithms:

6^x=90

x=2.51

Random Logarithm

100

State the Transformations:

log(x+3)

x+3: translates left

200

Solve with a calculator

3^(2x) = 2^(x-7)

x=-3.226

300

Solve without a calculator:

4^(x+3) = 32^(x-3)

x=7

400

State the transformations:

-ln(x-4)+1

-: reflection over the x-axis

x-4: right 4

+1: up 1

500

What is the linear equation that will represent this equation on a semilog plot with a scale of 10:

y = 75(1.7)^x

y' = 1.875 + 0.230x