R1 – Logs

This post covers the definition of R1 logs with details of the logarithmic formulae and 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 two numbers together by just adding the logs.

In classical maths, (-3)³ = -27 and (-3)⁴ = 81. However, logs only apply to positive numbers so you cannot say that log_-3(-27) = 3 or log_-3(81) = 4. Logs are only the inverse function to exponentiation in certain circumstances.

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^{^}

Logs are ^ Only

Only ^{^} Opposite Value bases and Opposite Values can use logs because R1 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.

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 and these are called the binary logs.  Write these logs as log_2^{^}(x), lb^{^} x, log₂(x) or lbx.

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

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

Logarithm Formulae

Logarithms facilitate multiplication because adding the logs of two terms equates to multiplication. 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^log_ab                    = 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

Conclusion

Our online calculator does not yet support logs.

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