Algebra

Logarithms

log_𝑏=𝑥 signifies that 𝑎^𝑥=𝑁, or Def.: The logarithm of a number is the power to which the base must be raised to produce that number.

Rules

log_𝑎𝑎=1, log1=0

Properties

log𝑀𝑁=log𝑀+log𝑁 log𝑀𝑁=log𝑀−log𝑁 log𝑀^𝑛=𝑛log𝑀 log^𝑛𝑀=1𝑛log𝑀 log_𝑏𝑎=log_𝑐𝑎log_𝑐𝑏 That is, the logarithm of a number of any base is equal to the logarithm of the number divided by the logarithm of the base, the two last named logarithms being taken to any the same base at pleasure.

Proof

Let log_𝑐𝑎=𝑥 and log_𝑐𝑏=𝑦; then 𝑎=𝑐^𝑥, 𝑏=𝑐^𝑦.
Eliminate 𝑐, 𝑐=𝑎^1𝑥=𝑏^1𝑦 ∴𝑏^𝑥𝑦, that is, log_𝑏𝑎=𝑥𝑦 log_𝑏𝑎=1log_𝑎𝑏 log₁₀𝑁=log_𝑒𝑁log_𝑒10 1log_𝑒10=0.43429448⋯ is called the modulus of the common system of logarithms; that is, the factor which will convert logarithms of numbers calculated to the base 𝑒 into the corresponding logarithms to the base 10.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive