R2 – Logs

This post covers the definition of R2 logs, logarithm formulae and provides some examples.

Definition

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.

The notation used for logs in Wave Numbers is log_b(x), which represents the log of x to the base b. A base must be a ^{^} 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:

• Log_10^{^}(1000^{^}) = 3
• Log_10^{^}(1^{^}/10) = ^–1

Example

• 1000^{^}  * 1^{^}/10  = 10^{^3} * 10^{^-1} = 10^{^2} = 100^{^}
  • Log_10^{^}(1000^{^}) + Log_10^{^}(1^{^}/10) = 3 + ^–1 = 2
  • 10^{^2} = 100^{^}

R2 Logs are ^ Only

Only ^{^} Opposite Value bases and Opposite Values can use logs because R2 logs for odd powers are not calculable for numbers such as log₃^v(81^v). This is because 3^v*3^v*3^v*3^v = 81^{^}. The base 3^v cannot be raised to 81^v by multiplying with itself.

Logs for the i^{^} Opposite Values are not always calculable such as for log_3i^{^}(9^{^}). No matter how often 3i^{^} is multiplied with itself, it will never result in 9^{^} as i^{^}*i^{^} = 1^v. The same applies to the i^v Opposite Values.

Other Bases

Typically the base of 10^{^}  is used and is known as the common log. Write this as log_10^{^}(x),  lg^{^} x, log₁₀(x) or lgx. Computers use the bases 2^{^} and 2 a lot. These are called the binary logs.  Write these as log_2^{^}(x), lb^{^} x, log₂(x) or lbx.

The base e^{^}  (Euler’s numbers) is the natural log. Write as log_e^{^}(x), ln^{^} x, log_e(x) or lnx.

It is not possible to add logs of different bases together.

R2 Logarithm Formulae

Logarithms facilitate multiplication because the R2 logs of two terms that require multiplication can add together to get the logarithm of the result. The following are the four basic logarithm formulae.

• Product:  log_b (xy)   = log_b(x) + log_b(y)
• Quotient: log_b (x/y)  = log_b(x) + ^–log_b(y)
• Power:     log_b(x^p)     = plog_b(x)
• Root:       log_b^p√x     = (log_b(x))/p)

Further Examples

• log_10^{^}(1^{^})              = 0

• log_c^{^}(c^{^})                = 1

• log_a^{^}b^{^}                  = log_c^{^}b/log_c^{^}a    For example:
  • log_10^{^}100^{^} = 2 = log_2^{^}100^{^}/log_2^{^}10
  • = 6.644/3.322 = 2

• log_a^{^}a^b                  = b  For example:
  • log_10^{^103}  = log_10^{^}1000  = 3

• a^logab                    = b  For example:
  • 10^{^log_10^1000_^} = 10^{^3} = 1000^{^} 
  • Here ^{log_10^1000_^} is the log of 1000 to the base 10^{^} = 3, so 10^{^3} = 1000^

• log_10^{^}(1000^{^})       = 3      

• log_10^{^}(100^{^})         = 2

• log_2^{^}(16^{^})             = 4

• log_10^{^}(10^{^}*100^{^}) = log_10^{^}(10^{^}) + log_10^{^}(100^{^})
  • = 1 + 2 = 3 = log_10^{^}(1000^{^}) = 3

• log_10^{^}(1000^{^}/10^{^}) = log_10^{^}(1000^{^}) + ^–log_10^{^}(10^{^})
  • = 3 + ^–1 = 2 = log_10^{^}(100^{^}) = 2

• log_10^{^}(1^{^}/10) = log_10^{^}(1^{^}) + ^–log_10^{^}(10^{^})
  • = 0 + ^–1 = ^–1

• log_10^{^}((1^{^}/10^{^})*100^{^}) = log_10^{^} (1^{^}/10^{^}) + log_10^{^}(100^{^})
  • = ^–1 + 2 = 1 = log_10^{^}(10^{^}) = 1

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