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Mastering Exponential and Logarithm Equations in Precalculus

Unlock the secrets of exponential and logarithmic equations with Dr. Claude S. Moore at Danville Community College. This comprehensive guide covers the properties of logarithms and exponentials, including inverse relationships and one-to-one properties. Learn how to solve exponential equations by isolating expressions and applying logarithms, as well as how to rewrite logarithmic equations as exponential forms. Through clear examples and step-by-step explanations, you'll grasp essential concepts to succeed in precalculus. Study diligently, and sky's the limit!

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Mastering Exponential and Logarithm Equations in Precalculus

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  1. PRECALCULUS I EXPONENTIAL & LOG EQUATIONS Dr. Claude S. MooreDanville Community College

  2. EXPONENTIAL & LOG INVERSE PROPERTIES 1. loga ax= x ln ex= x . 2. a log a x = x e lnx = x .

  3. ONE-TO-ONE PROPERTIES 1. loga x = loga y iff x = y. 2. a x= a y iff x = y. 3. a x= b x iff a = b.

  4. TO SOLVE ... • Exponential equation: Isolate exponential expression; take log of both sides and solve. • Logarithm equation: Rewrite as exponential equation and solve.

  5. EXAMPLE 1 • Solve (no calculator): 3x-1 = 243 3x-1 = 35 • Thus, x - 1 = 5 or x = 6.

  6. EXAMPLE 2 • Solve (3-decimal places): 4e2x = 50e2x = 50/4 = 12.5 • ln e2x = 2x(ln e) = ln 12.5 • x = (ln 12.5)/2 = 1.263

  7. EXAMPLE 3 Solve (3-decimal places):ln (x-2) + ln (2x+3) = ln x2 ln (x-2)(2x+3) = ln x2 ln (2x2-x-6) = ln x2 2x2-x-6 = x2

  8. EXAMPLE 3 continued 2x2-x-6 = x2 x2-x-6 = 0 (x-3)(x+2) = 0 x-3 = 0 or x+2 = 0 x = 3 or x = -2

  9. EXAMPLE 3 concluded ln (x-2) + ln (2x+3) = ln x2 Domain x-2>0, yields x>2 2x+3 > 0, yields x > -3/2 x2 > 0, yields x  0. So answer is x = 3.

  10. STUDY AND WORK HARD... ...the sky is the limit!

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