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Intro to Logarithms Definition A logarithm base b of a positive number x satisfies the following definition. For 𝑏 > 0, 𝑏 ≠ 1, 𝑙𝑜𝑔𝑏 𝑥 = 𝑦 if and only if 𝑏 𝑦 = 𝑥. You can read 𝑙𝑜𝑔𝑏 𝑥 as “log base b of x.” In other words, the logarithm y is the exponent to which b must be raised to get x. Different Bases Logs may be written three different ways, depending on the given base: = 𝑦: used for a log with a given base # 𝑙𝑜𝑔𝑥 = 𝑦: called the common log and used for base 10 (no base showing) 𝑙𝑛𝑥 = 𝑦: called the natural log and used for base e (spelled differently) 𝑙𝑜𝑔𝑏 𝑥 Exponent to Log form. Exponential Form: 𝑦 = 𝑏𝑥, 𝑏 > 0 Log Form: 𝑙𝑜𝑔𝑏 𝑦 = 𝑥 , 𝑏 > 0 Change from log to exponential form log 81 0 log 1000 3 1 log 2 4 16 Changing Forms Change from exponential to log form 2x 8 y 1 4 1 2 9 3 Evaluate log 2 64 5 log 4 n 2 Solving log equations by switching forms log 2 x 3 log 27 3 x Solving log equations by switching forms 1 log x 4 2 1 log 5 x 25 Log is the INVERSE of Exponential Characteristics of Inverses 1. Swap domain and range 2. Graphs are reflections across the line 𝑦=𝑥 Consider: 𝑦 = 2𝑥 and 𝑦 = 𝑙𝑜𝑔2 𝑥 Look at the graphs 𝑦 = 2𝑥 𝑦 = 𝑙𝑜𝑔2 𝑥 Parent Graph of the Log Function y log(x) Graph of a Log function y log( x 1) Graph of a Log function y log( x) 2 Graph of a Log function y log( x 3) 1 Graph of a Log function y log(x) Graph of a Log function y log( x) Graph of a Log function y log( x) 3