## Logarithms – Logs

Logarithms (or logs) define the decibel (dB) scale that is widely used in RF and microwave engineering

The logarithm is a type of scientific notation that defines the decibel (dB). The decibel scale is widely used in RF and microwave engineering, so a basic knowledge of logs is essential. But why are logs so useful, particularly when analysing the performance of wireless communications systems?

When RF signals are transmitted in free space, the magnitude of the radiating signal reduces rapidly as the distance from the transmitter increases. Consequently, the magnitude of the signal collected by a receiving antenna is normally many thousands or even millions of times smaller than the original transmission.

Using the everyday decimal number system in combination with signal powers measured in milliwatt (mW) units means that the transmitted signal levels will be very large numbers, with many zeros before the decimal point. Likewise, the received signal levels will be very small numbers, with many zeros after the decimal point.

Since many zeros are involved, calculations can be cumbersome and error-prone. It’s very easy to miss out a zero or even place a decimal point in the wrong place. Fortunately, logarithms make the calculation much simpler!

Before explaining how logs are used, it’s helpful to define what a logarithm is:

The logarithm of Y to the base X is the power L to which X must be raised to give Y

$$Y = X^L$$

In the conventional decimal (or base 10) number system, X = 10 and so

$$Y = 10^L$$

Some examples of base 10 logarithms are shown in the following table:

Y Notation L
10 101 1
100 102 2
1,000 103 3
10,000 104 4
0.1 10-1 -1
0.01 10-2 -2
0.001 10-3 -3
0.0001 10-4 -4
1 100 0

Using 1000 as an example:

$$1000 = 10^3$$

and therefore the log of 1000 is:

$$\log(1000) = 3$$

### Multiplying Numbers by Adding Logs

Logs make multiplying two numbers with the same base very easy. In the conventional decimal number system, the numbers are simply written down in base 10 and their logs added. For example:

$$10 \times 10000 =(10^1) \times (10^4) = 10^{1+4} = 10^5 = 100000$$

### Dividing Numbers by Subtracting Logs

Similarly, to divide two numbers with the same base (in this case base 10), subtract the log of the denominator from the log of the numerator. For example:

$$10000/10 = \displaystyle \frac{10^4}{10^1} = 10^{4-1} = 1000$$

Logarithms are used to define the bel and decibel scales. The decibel (dB) and its closely related measure of power the decibel-milliwatt (dBm), are widely used in RF and microwave engineering. Both are key performance metrics of many RF and microwave components and systems.