Solving Exponential and Logarithmic Equations Section 4.4 JMerrill, 2005 Revised, 2008

Same Base SSSSolve: 4x-2 = 64x  4 x-2 = (43)x x-2 = 43x  x –2 = 3x  - 2 = 2x 1 = x If b M = b N, then M = N 64 = 4 3 If the bases are already =, just solve the exponents

You Do  Solve 27 x+3 = 9 x-1

Review – Change Logs to Exponents llllog3x = 2 llllogx16 = 2 llllog 1000 = x 3 2 = x, x = 9 x 2 = 16, x = 4 10 x = 1000, x = 3

Using Properties to Solve Logarithmic Equations IIIIf the exponent is a variable, then take the natural log of both sides of the equation and use the appropriate property. TTTThen solve for the variable.

Example: Solving  2 x = 7 problem lllln2x = ln7 take ln both sides xxxxln2 = ln7power rule  x = divide to solve for x = 2.807

Example: Solving  e x = 72problem  l nex = ln 72take ln both sides  x lne = ln 72power rule = 4.277solution: because ln e = ?

You Do: Solving  2 ex + 8 = 20problem ex = 12subtract 8  e x = 6divide by 2  l n ex = ln 6take ln both sides  x lne = 1.792power rule x = 1.792(remember: lne = 1)

Example SSSSolve 5x-2 = 42x+3 lllln5x-2 = ln42x+3 ((((x-2)ln5 = (2x+3)ln4 TTTThe book wants you to distribute… IIIInstead, divide by ln4 ((((x-2) = 2x+3 1111.1609x = 2x+3 xxxx≈6.3424

Solving by Rewriting as an Exponential SSSSolve log4(x+3) = 2 44442 = x+3 11116 = x+3 11113 = x

You Do SSSSolve 3ln(2x) = 12 lllln(2x) = 4 RRRRealize that our base is e, so eeee4 = 2x xxxx ≈ YYYYou always need to check your answers because sometimes they don’t work!

Using Properties to Solve Logarithmic Equations 1111.Condense both sides first (if necessary). 2222.If the bases are the same on both sides, you can cancel the logs on both sides. 3333.Solve the simple equation

Example: Solve for x llllog36 = log33 + log3xproblem llllog36 = log33xcondense  6 = 3xdrop logs  2 = xsolution

You Do: Solve for x llllog 16 = x log 2problem llllog 16 = log 2xcondense  1 6 = 2xdrop logs  x = 4solution

You Do: Solve for x  l og4x = log44problem  = log44condense 4drop logs ccccube each side  X = 64solution

Example  7xlog 2 5 = 3xlog ½ log 2 25  log 2 5 7x = log 2 5 3x + log 2 25 ½  log 2 5 7x = log 2 5 3x + log  7x = 3x + 1  4x = 1

You Do  Solve:log log 7 2 = log 7 x + log 7 (5x – 3)

You Do Answer  Solve:log log 7 2 = log 7 x + log 7 (5x – 3)  log 7 14 = log 7 x(5x – 3)  14 = 5x 2 -3x  0 = 5x 2 – 3x – 14  0 = (5x + 7)(x – 2)  Do both answers work? NO!!

Final Example  How long will it take for $25,000 to grow to $500,000 at 9% annual interest compounded monthly?

Example