Definition
This post covers the definition of R3 logs, logarithm formulae and provides some examples.
Wikipedia defines logs as follows:
‘In mathematics, logarithm (log) is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.’
Logs are useful in expressing large and small numbers. They also provide a quick way to multiply 2 numbers together by just adding the logs.
Logs apply to all Opposite Types in R3.
The notation used for logs in Wave Numbers is logb(x), which represents the log of x to the base b. A base can be any Opposite Value.
Counters are used to represent the exponent to which a base is raised to get an Opposite Value. Flipped Counters are used to represent the exponent to which the reciprocal of a base is raised to get an Opposite Value that is a fraction. For example:
- Log10j^(1000j^) = 3
- Log10j^(j^/10) = –1
- 1000i^ * i^/10 = 103i^ * 10-1i^ = 102i^ = 100i^
- Log10i^(1000i^) + Log10i^(1i^/10) = 3 + –1 = 2
- 102i^ = 100i^
Other Bases
When the base of 10?^ is used, it is known as the common log. Write this as log10?^(x), lg?^ x, log10?(x) or lg?x.
However, computers use the bases 2^ and 2 a lot. These are called the binary logs. Write these as log2^(x), lb^ x, log2(x) or lbx.
The base e^ (Euler’s numbers) is called the natural log. Write as loge^(x), ln^ x, loge(x) or lnx.
Similar bases are available for each Opposite Type and Sign. It is not possible to add logs of different bases together.
R3 Log Formulae
R3 Logs facilitate multiplication because the logs of two terms that require multiplication can be added together to get logarithm of the result. The following are the four basic logarithm formulae.
- Product: logb (xy) = logb(x) + logb(y)
- Quotient: logb (x/y) = logb(x) + –logb(y)
- Power: logb(xp) = plogb(x)
- Root: logbp√x = (logb(x))/p)
Further Examples
- log10i^(i^) = 0
- logc(c) = 1
- loga(b) = logc(b)/logc(a) For example:
- log10j^(100j^) = 2 = log2j^(100j^)/log2j^(10j^)
- = 6.644/3.322 = 2
- loga(ab) = b For example:
- log10i^(103i^) = log10i^(1000i^) = 3
- aloga(b) = b For example:
- 10^log10^(1000^) = 103^ = 1000^
- Here log10^(1000^) is the log of 1000^ to the base 10^ = 3, so 103^ = 1000^
- log10j^(1000j^) = 3
- log10i^(100i^) = 2
- log2^(16^) = 4
- log10^(10^ * 100^ ) = log10^(10^) + log10^(100^)
- = 1 + 2 = 3 = log10^(1000^) = 3
- log10i^(1000i^/10i^) = log10i^(1000i^) + –log10i^(10i^)
- = 3 + –1 = 2
- = log10i^(100i^) = 2
- log10^(1^/10) = log10^(1^) + –log10^(10^) = 0 + –1 = –1
- log10i^((1i^/10i^)*100i^) = log10i^(1i^/10i^) + log10i^(100i^)
- = –1 + 2 = 1 = log10i^(10i^) = 1
- log10j^103j^ = log10j^1000j^ = 3
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