## Laws of logarithms (Engineering Mathematics)

In engineering mathematics there are seven laws of logarithms, which apply to logarithms in any base (including ln which is in base e). These laws are used in simplifying expressions involving logarithms and in solving equations involving logarithms. Lets us look at these laws in more detail.

### Law 1

log AB = log A + log B

Or

ln AB = ln A + ln B

### Law 2

$latex log \frac{A}{B} = log A – log B$

Or

$latex ln \frac{A}{B} = ln A – ln B$

### Law 3

log A^{b} = b log A

Or

ln A^{b} = b ln A

### Law 4

log_{B} B = 1

Or

log_{e} e = 1

ln e = 1

### Law 5

log 1 = 0

Or

ln 1 = 0

### Law 6

log_{A} B = $latex \frac{log B}{log A}$

### Law 7

log_{A} B = $latex \frac{1}{log_B A}$

## Simplifying logarithmic expressions using the laws of logarithms

Here are a few examples on simplifying logarithmic expressions using the laws of logarithms. These examples will help you understand how the laws of logarithms work.

### Example 1

Without using a calculator or mathematical tables, simplify the following as far as possible:

- log
_{5}23 + log_{5}2 - log
_{10}5 + log_{10}2 - log
_{6}87 – log_{6}3 - log
_{2}16 – log_{2}2 - ln 5 + ln 6
- log
_{10}64 − log_{10}128 + log_{10}32

### Solution 1

log_{5} 23 + log_{5} 2

*using law 1*- log
_{5}23 + log_{5}2 = log_{5}(23 × 2) - = log
_{10}46 *we cant simplify it any further than this***∴ log**_{5}23 + log_{5}2 = log_{10}46

log_{10} 5 + log_{10} 2

*using law 1*- log
_{10}5 + log_{10}2 = log_{10}(5 × 2) - = log
_{10}10 *using law 4*- log
_{10}10 = 1 **∴ log**_{10}5 + log_{10}2 = 1

log_{6} 87 – log_{6} 3

*using law 2*- log
_{6}88 – log_{6}2 = log_{6}$latex \frac{88}{2}$ - = log
_{6}44 *there is no law that can simplify it further than this***∴ log**_{6}87 – log_{6}3 = log_{6}44

log_{2}16 – log_{2}2

*using law 2*- log
_{2}16 – log_{2}2 = log_{2}$latex \frac{16}{2}$ - = log
_{2}8 *using the laws of indices*- = log
_{2}2^{3} *using law 3*- log
_{2}2^{3}= 3log_{2}2 *using law 4*- 3log
_{2}2 = 3 × 1 **∴ log**_{2}16 – log_{2}2 = 3

ln 6 + ln 5

*using law 1*- ln 6 + ln 5 = ln (6 × 5)
**∴ ln 6 + ln 5 = ln 30**

log_{10} 64 − log_{10} 128 + log_{10} 32

*first convert 64, 128 and 32 to indices*- log
_{10}2^{6}− log_{10}2^{7}+ log_{10}2^{5} *using law 3*- 6log
_{10}2 − 7log_{10}2 + 5log_{10}2 *factorising*- (6 – 7 + 5)log
_{10}2 **4log**_{10}2

## Solving logarithmic equations using the laws of logarithms

### Example 2

Using the laws of logarithms, solve the following equations:

- log x
^{4}− log x^{3}= log 5x − log 2x - log 2t
^{3}− log t = log 16 + log t

### Solution 2

log x^{4} − log x^{3} = log 5x − log 2x

*using law 2 on both the LHS and RHS of the equation*- log $latex \frac{x^4}{x^3}$ = log $latex \frac{5x}{2x}$
*simplify LHS using laws of indices and using basic algebra*- log x = log 2.5
*take antilogs of both sides***∴ x = 2.5**

log t^{3} − log t = log 16 + log t

*using law 2 on the LHS and law 1 on the RHS*- log $latex \frac{t^3}{t}$ = log 16t
- log t
^{2}= log 16t *take antilogs of both sides*- t
^{2}= 16t *solve the resulting quadratic equation*- t
^{2}= 16t - t
^{2}– 16t = 0 - t(t – 16) = 0
**∴ t = 0 or t = 16**