Logarithms

Converting Log and Exponential Form

100

Convert to Exponential Form:

log_2(8)=x

2^x=8

200

Convert to Logarithmic Form:

4^y=x

log_4(x)=y

300

Convert to Exponential Form:

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

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

400

Convert to Exponential Form:

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

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

or

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

500

Convert to Logarithmic Form:

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

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

Solving Log Equations with Log Properties

100

Solve for x:

log_3(x-3)=log_3(55)

x=58

200

Solve for x:

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

x=3

300

Solve for x:

log(2)+log(x)=log(x+12)

x=12

400

Solve for x:

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

x=32

500

Solve for x:

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

x=4

-1 is an extraneous solution

Expanding/Condensing Logarithms

100

Condense the Logarithms:

log_3(2x)-log_3(5y)

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

200

Completely Expand the Logarithm:

log((2x)/y)

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

300

Condense the Logarithms:

2log_4(3)-2log_4(x)+4log_4(y)

log_4((9y^4)/x^2)

400

Condense the Logarithms:

3log_5(x)-4log_5(2)-5log_5(z)+3log_5(3)

log_5((27x^3)/(16z^5))

500

Completely Expand the Logarithm:

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

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

Solving Exponential Equations with Logs (calculator)

100

Solve using Logarithms. Round the answer to the hundredth if necessary:

6^x=90

x=2.51

200

Solve Using Logarithms. Round the answer to the hundredth if necessary:

3^(7x)=108

x=0.61

300

Solve by Converting:

log_4(x)=5

x=1024

400

Solve using Logarithms. Round the answer to the hundredth if necessary:

4^(x/3)=60

x=8.86

500

Solve using Logarithms. Round the answer to the hundredth if necessary:

4^(x-2)=100

x=5.32

Foundational logs

100

log_xx

1

200

log₂₄1

0

300

log_K(-16)=

Undefined  

400

log 100

2

500

log_7x(7x)⁵

5