LOGARITHMS!

exponential form to Log form

100

2⁴=16

log₂(16)=4

200

0.001=100^-3/2

log₁₀₀(0.001)=-3/2

300

9⁰=1

log₉(1)=0

400

16^1/4=2

log₁₆(2)=1/4

500

625^3/4=125

log₆₂₅(125)=3/4

Log form to exponential form

100

log₁₀(0.1)=-1

10^-1=0.1

200

log₁₄(1/196)=-2

14^-2=1/196

300

log_v(u)=4

v⁴=u

400

log₂₇(9)=2/3

27^2/3=9

500

log_7/4(x)=y

7/4^y=x

Solving equation for variable

100

log x +log 8 = log 200

log x +log 8 = log 200

log (8x) = 200 

8x=200 

x=25

200

log x + log 18=log 27

log x + log 18=log 27

log (18x)=27

x=1.5

300

log(x+3)=log(x-5)+log(3)

log (x+3)=log (x-5)+log (3)

log (x+3)=log ((3)(x-5))

x+3=3x-15

x=9

400

log₂(2³) +log₂(2²⁾=log₂(x)

log₂(2³) +log₂(2²⁾=log₂(x)

3+2=log₂(x)      log₂(2⁵)=log₂(x)

5 = log₂(x) 

x=2⁵ = 32            x=32

500

log₂(8)+log₃(9)=log_b(100,000)

log₂(8)+log₃(9)=log_b(100,000)

3+2=log₅(100,000) 

b⁵=100,000 

b=10 

solving equations with the same base

100

2^x=16

2^x=16

2x2x2x2=16 

2⁴=16 

x=4

200

g^2x+2=g^x-1

g^2x+2=g^x-1

2x+2=x-1

x=-3

300

3^x+1=3^2x+3

3^x+1=3^2x+3

x+1=2x+3

x=2

400

49^x=7^3x+1

49^x=7^3x+1

49^x=7^2x  =  7^2x=7^3x+1 

2x=3x+1

x=-1

500

5^x-1=(0.04)^2x

5^x-1=(0.04)^2x

5^x-1=(4/100)^2x    =   5^x-1=(1/25)^2x    =  5^x-1=(5^-2)^2x

x-1=-4x

x=1/5

Random Problems (simplify, expand, etc.)

100

Write the expression as a single logarithm

log y -log (y-1)

log y - log(y-1)

log (y/y-1) 

200

Expand log₃(10/x)

log₃(10/x)

log₃(10)-log₃(x) 

300

Expand log₄(x⁶/y⁵)

log₄(x⁶/y⁵)

log₄(x⁶)-log₄(y⁵)

6log₄(x) - 5log₄(y)

400

log₃(27)+log₃(81)

log₃(27x81)=log₃(3³x3⁴)

                 =log₃(3⁷)=7

500

log₅(25)+2log₁₀(10)

________________

        log₁₆(4)

(this is a fraction) 

log₅(25)=2

log₁₀(10)=1

2+2(1)            

_______  =  4/(1/2)=8 

    1/2