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Solving Exponential & Logarithmic Equations. Strategies and Practice. Exponentials & Equal Bases. Equal bases must have equal exponents EX: Given 3 x-1 = 3 2x + 1 then x-1 = 2x+1 x = -2 If possible, rewrite to make bases equal EX: Given 2 -x = 4 x+1 rewrite 4 as 2 2
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Solving Exponential & Logarithmic Equations Strategies and Practice
Exponentials & Equal Bases Equal bases must have equal exponents EX: Given 3x-1 = 32x + 1thenx-1 = 2x+1 x = -2 If possible, rewrite to make bases equal EX: Given 2-x = 4x+1 rewrite 4 as 22 2-x = 22x+2 then –x=2x+2 x=-2/3
Exponentials of Unequal Bases Use logarithm (inverse function) of same base on both sides of equation EX: Solve: ex = 72 lnex = ln72 xlne = ln72 x = ln72 (calc ready form) x ~ 4.277 EX: Solve: 7x-1 = 12 log77x-1 = log712 (x-1)log77 = log712 x-1 = log712 x = 1+log712 x ~ 1.277 You try… Solve e2x = 5
Single Side Log Equations Convert to exponential (inverse) form EX: Solve: lnx = -1/2 e-1/2 = x .607 ~ x EX: Solve: 2log53x = 4 log53x = 2 52 = 3x 25/3 = x Use Laws to condense EX: Solve: log4x – log4(x-1) = ½ log4(x2-x)= ½ 41/2 = x2 – x 0 = x2-x-2 (x-2)(x+1) x=2 WHY NOT -1? You try… lnx = -7
Double-Sided Log Equations Equate powers EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x + 7 x = 2 EX: Solve: ln(x-2) + ln(2x-3) = 2lnx Use a property:ln(x-2)(2x-3) = lnx2 2x2 – 7x + 6 = x2 x2-7x+6=0 x = 6 & 1 You try… Solve ln3x2 = lnx
SUMMARY • Equal bases Equal exponents • Unequal bases Apply log of given base • Single side logs Convert to exp form • Double-sided logs Equate powers Note: Any solutions that result in a log(neg) cannot be used!
