Logarithms

An exponential number is a function of the form 𝑏^𝑛 where 𝑏 is known as the base and 𝑛 is the exponent, power, or index of the exponentiation. In general, logarithms are exponents. If a number, 𝑥, is expressed in form of exponential number, 𝑏^𝑦, then the logarithm with base 𝑏 of the number, 𝑥, is equal to 𝑦, the exponent of the exponential number, 𝑥. 𝑥=𝑏𝑦⇒log𝑏𝑥=𝑦 The definition of logarithm is Definiton (Logarithm) Logarithm is a function. The logarithm of a numebr, 𝑥, is defined as the power to which a given base, 𝑏, must be raised in order to produce that number. Provided that 𝑥>0, and 𝑏 is any number such that 𝑏>0 and 𝑏≠1. 𝑦=log𝑏𝑥⇒𝑥=𝑏𝑦

Systems of Logarithms

A system of logarithms can be produced for a specific base by raising to various powers. The two most common logarithm systems are common logarithms and natural logarithems.

Common Logarithms

Logarithms having a base of 10 are called common logarithms. The common logarithm function is usually denoted by log₁₀ and is usually abbreviated to lg or log. For example, log₁₀ 10=lg 10=log 10=1.

Natural Logarithms

Logarithms having a base of ℯ are called natural, hyperbolic, or Napierian logarithms. The natural logarithm function is usually denoted by log_ℯ and is usually abbreviated to ln. For example, log_ℯ ℯ=ln ℯ=1.

Laws of Logarithms

Logarithms aid in multiplying, dividing, and raising numbers to higher powers. There are three laws of logarithms, which apply to any base:

• Law of Multiplication: Product Rule log (𝐴×𝐵)=log 𝐴+log 𝐵 To multiply 𝐴 by 𝐵, the log of 𝐵 is added to the log of 𝐴.
• Law of Division: Quotient Rule log 𝐴𝐵=log 𝐴−log 𝐵 To divide 𝐴 by 𝐵, the log of 𝐵 is subtracted from the log of 𝐴.
• Law of Raising Power: Power Rule log 𝐴𝑛=𝑛log 𝐴 To raise a number to a higher power, the log is multiplied by the power indicator. To extract the root of a number, the log is divided by the root indicator.

Properties of Logarithms

• log_𝑏1=0. ∵𝑏0=1⇒0=log𝑏1
• log_𝑏𝑏=1. ∵𝑏1=𝑏⇒1=log𝑏𝑏
• log_𝑏𝑏^𝑥=𝑥. Let 𝑏𝑦=𝑏𝑥⇒𝑦=𝑥 ∴𝑦=log𝑏𝑏𝑥 ⇒𝑥=log𝑏𝑏𝑥 generalized to ⇒log_𝑏𝑏^𝑓(𝑥)=𝑓(𝑥) Let 𝑓(𝑝)=𝑏𝑝; 𝑔(𝑞)=log𝑏𝑞 (𝑔∘𝑓)(𝑝)=𝑔(𝑓(𝑝))=𝑔(𝑏𝑝)=log𝑏𝑏𝑝=𝑝
• 𝑏^log_𝑏𝑥=𝑥. Let 𝑏𝑦=𝑥 ∴𝑦=log𝑏𝑥 ⇒𝑏log𝑏𝑥=𝑥 generalized to ⇒𝑏^{log_𝑏𝑓(𝑥)}=𝑓(𝑥) Let 𝑓(𝑝)=𝑏𝑝; 𝑔(𝑞)=log𝑏𝑞 (𝑓∘𝑔)(𝑞)=𝑓(𝑔(𝑞))=𝑓(log𝑏𝑥)=𝑏log𝑏𝑞=𝑞 Since (𝑔∘𝑓)(𝑝)=𝑝 and (𝑓∘𝑔)(𝑞)=𝑞, the exponential and logarithm functions are inverses of each other. Let 𝑓(𝑝)=𝑏𝑝; 𝑔(𝑞)=log𝑏𝑞 Assume 𝑓 is an inverse function for 𝑔. Let 𝑓(𝑝)=q, then 𝑔(𝑞)=𝑝 ⇒(𝑔∘𝑓)(𝑝)=𝑔(𝑓(𝑝))=𝑔(𝑞)=𝑝 ⇒(𝑓∘𝑔)(𝑞)=𝑓(𝑔(𝑞))=𝑓(𝑝)=𝑞 Conversely, 𝑔(𝑞)=𝑔(𝑓(𝑝))=𝑝 Conversely, 𝑓(𝑝)=𝑓(𝑔(𝑞))=𝑞
• log_𝑏(𝑥×𝑦)=log_𝑏𝑥+log_𝑏𝑦. Let 𝑥=𝑏𝑝⇒log𝑏 𝑥=𝑝 and 𝑦=𝑏𝑞⇒log𝑏 𝑦=𝑞 𝑥×𝑦=𝑏𝑝×𝑏𝑞=𝑏𝑝+𝑞 log𝑏(𝑥×𝑦)=log𝑏𝑏𝑝+𝑞 log𝑏(𝑥×𝑦)=(𝑝+𝑞)log𝑏𝑏 log𝑏(𝑥×𝑦)=𝑝+𝑞 log𝑏(𝑥×𝑦)=log𝑏𝑥+log𝑏𝑦
• log_𝑏𝑥𝑦=log_𝑏𝑥−log_𝑏𝑦 Let 𝑥=𝑏𝑝⇒log𝑏 𝑥=𝑝 and 𝑦=𝑏𝑞⇒log𝑏 𝑦=𝑞 𝑥𝑦=𝑏𝑝𝑏𝑞=𝑏𝑝−𝑞 log𝑏𝑥𝑦=log𝑏𝑏𝑝−𝑞 log𝑏𝑥𝑦=𝑝−𝑞 log𝑏𝑥𝑦=log𝑏 𝑥−log𝑏 𝑦
• log_𝑏𝑥^𝑛=𝑛log_𝑏𝑥 Let 𝑥=𝑏𝑝⇒log𝑏 𝑥=𝑝 𝑥𝑛=(𝑏𝑝)𝑛=𝑏𝑝𝑛 log𝑏𝑥𝑛=𝑝𝑛=𝑛𝑝 log𝑏𝑥𝑛=𝑛log𝑏 𝑥
• If log_𝑏𝑥=log_𝑏𝑦 then 𝑥=𝑦 Let 𝑝=log𝑏𝑥=log𝑏𝑦 𝑏𝑝=𝑏log𝑏𝑥=𝑏log𝑏𝑦 𝑏𝑝=𝑥=𝑦 ⇒𝑥=𝑦

Change of Base

Since most calculators are only capable of evaluating common logarithms and natural logarithms, method of change of base is needed to evaluate any other logarithms other than common logarithms and natural logarithms. The change of base formula is log𝑎𝑥=log𝑏𝑥log𝑏𝑎 if 𝑥=𝑏 ⇒log𝑎𝑏=log𝑏𝑏log𝑏𝑎=1log𝑏𝑎 Proof: Let 𝑝=log𝑎𝑥 ⇒𝑎𝑝=𝑎log𝑎𝑥=𝑥 ⇒logb𝑎𝑝=logb𝑥 ⇒𝑝logb𝑎=logb𝑥 ⇒𝑝=logb𝑥logb𝑎 ⇒log𝑎𝑥=logb𝑥logb𝑎 if 𝑥=𝑏 ⇒log𝑎𝑥=logb𝑏logb𝑎 ⇒log𝑎𝑥=1logb𝑎