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Solving Exponential & Logarithmic Equations

Learn effective strategies for solving exponential and logarithmic equations with step-by-step examples. This guide covers techniques for equal bases, unequal bases, single-sided logarithmic equations, and double-sided logarithmic equations. Understand how to rewrite equations to match bases, use logarithms for inverse functions, and apply properties of logarithms to condense and solve complex equations. Practice problems included will enhance your skills in this important area of mathematics.

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Solving Exponential & Logarithmic Equations

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  1. Solving Exponential & Logarithmic Equations Strategies and Practice

  2. 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

  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

  4. 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

  5. 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

  6. 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!

  7. Hw pg 311 5-80 mults of 5

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