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: 10^2.3010= 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=a^x then,

log_ay = 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:

3^p * 5^(p-2) = 8^2p * 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:

7.16^y = 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

log_ex = n

then eⁿ = x

n * log₁₀e = log₁₀x

n = log_ex = log₁₀x / log₁₀e

since 1 / log₁₀e = 2.3026

log_ex = log₁₀x * 2.3026

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