school 683556 640 Introduction to Logarithms (Engineering Mathematics)

Introduction to Logarithms (Engineering Mathematics)

Logarithms are a very important aspect of all engineering fields.

  • Chemical engineers measure radioactive decay using natural logarithms and also the pH of solutions is measured as the logarithm of the concentration of hydrogen ions.
  • Geologists and seismographers measure the intensity of earthquakes on a logarithmic scale called the Richter scale.
  • In biomedical engineering use cell decay and growth can be simulated using logarithms.
  • Electronic engineers and sound engineers measure amplification and sound levels using a logarithmic dB (decibel) scale.

These examples are to name but a few of the application of logarithms, but from these few examples we can see how important logarithms are in all branches of engineering.

Theory of logarithms


To fully understand what logarithms are, we first need to understand what indices are. Lets take for example:

y = bx

In the above expression, b is called the base and x is called the power or index or exponent.

We can alternatively rewrite the above expression as:

logby = x

which is pronounced as logarithm of y to base b is equal to x.

y = ax means that x = loga y

Theory of Logarithms Video Tutorial

Example 1

Rewrite the following indices in logarithm form:

  • 64 = 82
  • 4 = 160.5
  • x = 52

Solution 1

64 = 82

  • log864 = 2

4 = 160.5

  • log164 = 0.5

x = 52

  • log5x = 2

Common logarithms

In the previous section we learnt that if 1000 = 103, then log10 1000 = 3 which stands for "logarithm of 1000 to base 10 is equal to 3."

Since the most commonly used base in mathematics is base 10, logarithms having a base of 10 are called common logarithms and as such "log10" is usually abbreviated to just "lg".

Napierian logarithms

Logarithms to base of e (where e is Eula’s number, approximately equal to 2.7183) are called hyperbolic logarithms, Napierian logarithms or natural logarithms, referred to as loge, usually abbreviated to ln.

y = ex means that x = loge y or x = ln y

Simplifying logarithmic expressions

We can simplify logarithmic expressions by first converting them to indices and then simplifying the indices.

Example 2

Evaluate the following logarithms:

  • log10 1000
  • log2 8
  • log100 10
  • log16 2
  • ln e
  • ln e3

Solution 2

log10 1000

  • we first start by equating the expression to x
  • log10 1000 = x
  • then convert the resulting equation to index notation
  • 10x = 1000
  • express both sides of the equation to the same base
  • 10x = 103
  • if the bases are equal, then the powers are equal
  • x = 3
  • ∴ log10 1000 = 3

log2 8

  • log2 8 = x
  • 2x = 8
  • 10x = 23
  • x = 3
  • ∴ log2 8 = 3

log100 10

  • log100 10 = x
  • 100x = 10
  • 102x = 101
  • 2x = 1
  • x = 0.5
  • ∴ log100 10 = 0.5

log16 2

  • log16 2 = x
  • 16x = 2
  • 24x = 21
  • 4x = 1
  • x = 0.25
  • ∴ log16 2 = 0.25

ln e

  • ln e = x
  • loge e = x
  • ex = e
  • ex = e1
  • x = 1
  • ∴ ln e = 1

ln e3

  • ln e3 = x
  • loge e3 = x
  • ex = e3
  • ex = e3
  • x = 3
  • ∴ ln e3 = 3

Example 3

Solve the following equations:

  • log10 x = 4

Solution 3

log10 x = 4

  • first convert the equation to index notation
  • 104 = x
  • x = 104
  • x = 10 000