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3.4 Solving Exponential and Logarithmic Equations

3.4 Solving Exponential and Logarithmic Equations. Exponential Equations. One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. For b>0 & b ≠1 if b x = b y , then x=y. Solve by equating exponents.

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3.4 Solving Exponential and Logarithmic Equations

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  1. 3.4Solving Exponential and Logarithmic Equations

  2. Exponential Equations • One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal. • For b>0 & b≠1 if bx = by, then x=y

  3. Solve by equating exponents • 43x = 8x+1 • (22)3x = (23)x+1 rewrite w/ same base • 26x = 23x+3 • 6x = 3x+3 • x = 1 Check → 43*1 = 81+1 64 = 64

  4. Your turn! • 24x = 32x-1 • 24x = (25)x-1 • 4x = 5x-5 • 5 = x Be sure to check your answer!!!

  5. When you can’t rewrite using the same base, you can solve by taking a log or ln of both sides • 2x = 7 • log22x = log27 • x = log27 • x = ≈ 2.807

  6. 4x = 15 • log44x = log415 • x = log415 = log15/log4 • ≈ 1.95 Using Log Using LN

  7. 102x-3+4 = 21 • -4 -4 • 102x-3 = 17 • log10102x-3 = log1017 • 2x-3 = log 17 • 2x = 3 + log17 • x = ½(3 + log17) • ≈ 2.115

  8. Solving an Exponential Equation containing e • 40e0.6x-3= 237

  9. Solving Log Equations • To solve use the property for logs w/ the same base use the one-to one property: • If logbx = logby, then x = y

  10. log3(5x-1) = log3(x+7) • 5x – 1 = x + 7 • 5x = x + 8 • 4x = 8 • x = 2 and check • log3(5*2-1) = log3(2+7) • log39 = log39

  11. log5(3x + 1) = 2 Write in exponential form then solve! • 3x+1 = 25 x = 8 and check • Because the domain of log functions doesn’t include all reals, you should check for extraneous solutions

  12. log5x + log(x+1)=2 • log (5x)(x+1) = 2 (product property) • log (5x2 – 5x) = 2 • 5x2 - 5x = 100 • x2 – x - 20 = 0 (subtract 100 and divide by 5) • (x-5)(x+4) = 0 x=5, x=-4 • graph and you’ll see 5=x is the only solution

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