## 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

$latex ln \frac{A}{B} = ln A – ln B$

log Ab = b log A

Or

ln Ab = b ln A

logB B = 1

Or

loge e = 1

ln e = 1

log 1 = 0

Or

ln 1 = 0

### Law 6

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

### Law 7

logA 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:

• log5 23 + log5 2
• log10 5 + log10 2
• log6 87 – log6 3
• log216 – log22
• ln 5 + ln 6
• log10 64 − log10 128 + log10 32

### Solution 1

log5 23 + log5 2

• using law 1
• log5 23 + log5 2 = log5 (23 × 2)
• = log10 46
• we cant simplify it any further than this
• ∴ log5 23 + log5 2 = log10 46

log10 5 + log10 2

• using law 1
• log10 5 + log10 2 = log10 (5 × 2)
• = log10 10
• using law 4
• log10 10 = 1
• ∴ log10 5 + log10 2 = 1

log6 87 – log6 3

• using law 2
• log6 88 – log6 2 = log6 $latex \frac{88}{2}$
• = log6 44
• there is no law that can simplify it further than this
• ∴ log6 87 – log6 3 = log6 44

log216 – log22

• using law 2
• log216 – log22 = log2 $latex \frac{16}{2}$
• = log2 8
• using the laws of indices
• = log2 23
• using law 3
• log2 23 = 3log2 2
• using law 4
• 3log2 2 = 3 × 1
• ∴ log216 – log22 = 3

ln 6 + ln 5

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

log10 64 − log10 128 + log10 32

• first convert 64, 128 and 32 to indices
• log10 26 − log10 27 + log10 25
• using law 3
• 6log10 2 − 7log10 2 + 5log10 2
• factorising
• (6 – 7 + 5)log10 2
• 4log10 2

## Solving logarithmic equations using the laws of logarithms

### Example 2

Using the laws of logarithms, solve the following equations:

• log x4 − log x3 = log 5x − log 2x
• log 2t3 − log t = log 16 + log t

### Solution 2

log x4 − log x3 = 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 t3 − 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 t2 = log 16t
• take antilogs of both sides
• t2 = 16t
• solve the resulting quadratic equation
• t2 = 16t
• t2 – 16t = 0
• t(t – 16) = 0
• ∴ t = 0 or t = 16