Logarithms

Properties

100

log₅1=0

Property 1

200

log₄4=1

Property 2

300

log₂2⁷=7

Property 3

400

(5)^log₅⁷=7

Property 4

500

log₉9^(1/2)=1/2

Property 3

Conversions

100

log₃9=3

3³=9

200

log2=x

10^x=2

300

log₄20=x

4^x=20

400

log(3x²+4)=1

10=3x²+4

500

ln3⁴=15x

e^15x=3⁴

Real-life

100

The hydrogen ion concentration of a solution is 2 x 10⁴ mol/L. Calculate the pH.

pH = 3.7

200

The pH of the rainwater is 5.5. Calculate the hydrogen ion concentration.

3.16 x 10^-6

300

What is the magnitude of an earthquake if the amplitude of the vibrations is 500?

M = 2.7

400

What is the magnitude of an earthquake that has an intensity of 8.3?

M = 4.9

500

What is the intensity level of a thunderclap if the decibel level is measured at 120/dB?

1 W/m²

Natural Logs

100

ln(x)³

3 ln(x)

200

ln(ab)^1/2

1/2 ln(a) + 1/2 ln(b)

300

ln(x⁴ / 2)

4 ln(x) - ln(2)

400

ln(x(y/z)^1/2)

ln(x) +1/2 (ln(y) - ln(z))

500

ln(r^1/3/3s)

1/3 ln(r) - ln(3s)

Expansion/Condense

100

log(3x)

log(3) + log(x)

200

log₂(4xy)³

3 log₂(4) + 3 log₂(x) + 3 log₂(y)

300

2 log₄(x) - y log₄(3)

log₄(x²/3^y)

400

log₃((x²z)/y²)

2 log₃(x) + log₃(z) +2 log₃(y)

500

3 log₄(z) + 2 log₄(x) - 5 log₄(y)

log₄((z³x²)/y⁵)