Logarithms

Logarithms are used to simplify math. Multiplication and division can be accomplished through addition and subtraction.

 

log(1) = 0

log(number greater than 1) = positive

log(number less than 1) = negative

log(AB) = log(A) + log(B)

log(A/B) = log(A) - log(B)

log(A)n = nlog(A)

 

 

 

 

 

Power ratio expressed in dB

dB stands for decibel (1/10 of a Bel). The dB is a way of expressing the ratio between two power levels logarithmically.

 

 

P(dB) = Numerical power ratio expressed in dB

P1 = Power 1 (input)

P2 = Power 2 (output)

 

 

Decibels can be added and subtracted to determine the total gain of a system.

 

Pout = 10dBm + 5dB - 10dB + 20dB = 25dBm

 

 

P(dB) = 10 log10 (P2/P1)

Ratio

V(dB) = 20 log10 (V2/V1)

120

1.00E+12

240

90

1.00E+09

180

60

1.00E+06

120

50

1.00E+05

100

40

1.00E+04

80

30

1000

60

20

100

40

10

10

20

9.5

9

19.1

9.0

8

18.1

8.5

7

16.9

7.8

6

15.6

7.0

5

14.0

6.0

4

12.0

4.8

3

9.5

3.0

2

6.0

0

1

0

-0.5

0.9

-0.9

-1.0

0.8

-1.9

-1.5

0.7

-3.1

-2.2

0.6

-4.4

-3.0

0.5

-6.0

-4.0

0.4

-8.0

-5.2

0.3

-10.5

-7.0

0.2

-14.0

-10

0.1

-20

-20

0.01

-40

-30

0.001

-60

-40

1.00E-04

-80

-50

1.00E-05

-100

-60

1.00E-06

-120

-90

1.00E-09

-180

-120

1.00E-12

-240

 

 

 

Power expressed in dBm

Most of the time, power is expressed in dBm in RF applications. dBm is power relative to one milliwatt.

 

 

dBm Conversion Chart

 

 

 

 

Voltage ratio expressed in dB

As long as you use a consistent resistance you can express voltage as a logarithmic ratio as follows:

 

 

and

 

 

 

 

 

When R1 = R2 the second term goes to 0.

 

 

V(dB) = Numerical voltage ratio expressed in dB

V1 = Voltage 1 (input)

V2 = Voltage 2 (output)