# Maths - Logarithms

There are a number of uses for logarithms, often abbreviated to logs, one is that it is the inverse function to raising a number to a power. Another is that it can convert between multipication and addition, for instance,

log 3 + log 4 = log 12

## Logs

The 'log' of a number, to a given base, is the power to which the base must be raised to give the number.

e.g.,

if: log10200 = 2.3010

then: 102.3010= 200

Consider the base 2

since 23=8 then log28 = 3

since 24=16 then log216 = 4

in general if the base = a and y=ax then,

logay = x

## Some Properties of logs

 The log of the base itself is always unity since a1 = a then logaa = 1 The log of one is always zero since a0 = 1 then loga1 = 0 The log of a number is equal to minus the log of its reciprocal logax = - loga(1/x) loga(x * y) = logax + logay loga(x ÷ y) = logax - logay loga(x)n = n*logax logan√(x) = (1/n)*logax loga(x)n = n*logax

## Examples

1. 3√(0.0838) = antilog(-0.389/3)
2. (0.0273)2/3 = antilog(2/3 * log 0.0273) = 0.09067
3. (6.023)-2.5 = antilog(-2.5 * log 6.023) = antilog(-2.5 * 0.7798) = antilog(-1.9495) = 0.01123
4. (0.1276)-1.7 = antilog(-1.7 * log (0.1276) = antilog(-1.51997) = 33.11

## Solution of exponential equations

An exponential equation is one in which the unknown quantity is an index.

Example 1:

3p * 5(p-2) = 82p * 7(1-p)
p log 3 + (p-2) log 5 = 2p log 8 + (1-p) log 7
0.4771 p + 0.699(p-2) = 0.9031(2p) + 0.8451(1-p)
0.215 p = 2.2431
p=10.43

Example 2:

7.16y = 1.92 (y+2)
y log 7.16 = (y+2)log1.92
0.8549 y = 0.2833 (y+2)
0.8549 y - 0.2833y = 0.5666
y = 0.5666/0.5716

Example 3:

200 = k * 12 1.25* 80 -0.5
log 200 = log k + 1.25*log12 +(-0.5 log 80)
2.301 = log k + 1.25*1.0792 - 0.5*1.9031
log k = 1.90355
k = 80.1

## Bases other than 10

If 5 2= 25 then by definition log525 = 2

If 4 3= 64 then by definition log464 = 3

### Examples

evaluate log6216

let log6216 = x

then 6 x = 216

x * log106 = log10216

x = 2.3345/0.7782

x = 3

### Logs to the power of 2

evaluate log21.87

let log21.87 = x

then 2 x = 1.87

x * log102 = log101.87

x = 0.2718/0.301

x = 0.9241

## Natural or Naperian Logs

The base 'e' is very common base in engineering, where:

e = 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) ...

the value of e to 5 decimal places is = 2.71828

To change a log from base e to base 10

logex = n

then en = x

n * log10e = log10x

n = logex = log10x / log10e

since 1 / log10e = 2.3026

logex = log10x * 2.3026

 metadata block see also: Correspondence about this page Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.       The MathML Handbook - for people interested in working with mathematics on the web. Other Math Books Specific to this page here:

This site may have errors. Don't use for critical systems.