BDM, EAP 7/27/2016 Relationship between Logarithmic Functions & Exponential Functions DEFINITION: A logarithm is the inverse of an exponential function. EXAMPLE: Given: π(π) = π = ππ Find: ππ§π―ππ«π¬π π¨π π(π) π¨π« πβπ (π) Solution: π = ππ How do you solve for y? Use a logarithmic function: πβπ (π) = π = π₯π¨π π π Solving logarithmic functions: To solve logarithmic functions, one needs to know the following relationship between exponents and logarithms. FORMULAS: ππ = π βΊ π = π₯π¨π π π EXAMPLES: πππ = πππ βΊ π = π₯π¨π ππ πππ The rules for simplifying and manipulating logarithmic expressions somewhat resemble the rules for exponents. Operation Laws of Exponents Laws of Logarithms ππ = π π§ so π = π§ Identity ππππ π₯ = ππππ π¦ so π₯ = π¦ Identity π₯ =π₯ Division β Subtraction π₯π = π₯π β π₯ π§ π§ π₯ ππππ π = 1 (or ππππ π1 = 1) so ππππ π π₯ = π₯ π₯ ππππ ( ) = ππππ π₯ β ππππ π¦ π¦ Multiplication β Addition π₯ π β π₯ π§ = π₯ π+π§ ππππ ( π₯ β π¦) = ππππ π₯ + ππππ π¦ 1 (π₯ π )π = π₯ 3βπ = π₯ 3π ππππ (π¦ π )π = 3 β ππππ π¦1 = 3 log π π¦ Power of a Power (proved by Addition) Proof: (π₯ π )π = π₯ π β π₯ π β π₯ π = π₯ π+π+π = π₯3π Proof: ππππ ( π¦ π )π = ππππ( π¦ β π¦ β π¦) = ππππ π¦ + ππππ π¦ + ππππ π¦ = 3 ππππ π¦ The Change of base formula is useful because some calculators can only compute logarithms of base 10 (the common logarithm, often written simply as log) or base e (the natural log, ln, for which base e is understood). log π π¦ = πππ10 π¦ πππ π¦ ππ π¦ (also ) or πππ10 π πππ π ππ π Find handouts and links to other math resources at http://www.lsco.edu/learningcenter/math.asp The Learning Center @ LSC-O, Ron E. Lewis Library building, room 113, 409-882-3373
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