## 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 = b ^{x} `

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:

` log _{b}y = x `

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

y = a^{x} means that x = log_{a} y

### Example 1

Rewrite the following indices in logarithm form:

- 64 = 8
^{2} - 4 = 16
^{0.5} - x = 5
^{2}

### Solution 1

64 = 8^{2}

- log
_{8}64 = 2

4 = 16^{0.5}

- log
_{16}4 = 0.5

x = 5^{2}

- log
_{5}x = 2

### Common logarithms

In the previous section we learnt that if 1000 = 10^{3}, then log_{10} 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 "log_{10}" 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 log_{e}, usually abbreviated to ln.

y = e^{x} means that x = log_{e} 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:

- log
_{10}1000 - log
_{2}8 - log
_{100}10 - log
_{16}2 - ln e
- ln e
^{3}

### Solution 2

log_{10} 1000

*we first start by equating the expression to x*- log
_{10}1000 = x *then convert the resulting equation to index notation*- 10
^{x}= 1000 *express both sides of the equation to the same base*- 10
^{x}= 10^{3} *if the bases are equal, then the powers are equal*- x = 3
**∴ log**_{10}1000 = 3

log_{2} 8

- log
_{2}8 = x - 2
^{x}= 8 - 10
^{x}= 2^{3} - x = 3
**∴ log**_{2}8 = 3

log_{100} 10

- log
_{100}10 = x - 100
^{x}= 10 - 10
^{2x}= 10^{1} - 2x = 1
- x = 0.5
**∴ log**_{100}10 = 0.5

log_{16} 2

- log
_{16}2 = x - 16
^{x}= 2 - 2
^{4x}= 2^{1} - 4x = 1
- x = 0.25
**∴ log**_{16}2 = 0.25

ln e

- ln e = x
- log
_{e}e = x - e
^{x}= e - e
^{x}= e^{1} - x = 1
**∴ ln e = 1**

ln e^{3}

- ln e
^{3}= x - log
_{e}e^{3}= x - e
^{x}= e^{3} - e
^{x}= e^{3} - x = 3
**∴ ln e**^{3}= 3

### Example 3

Solve the following equations:

- log
_{10}x = 4

### Solution 3

log_{10} x = 4

*first convert the equation to index notation*- 10
^{4}= x - x = 10
^{4} **x = 10 000**