## 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

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