3.4 Solving Exponential and Logarithmic 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 Exponential Equations

Solve by equating exponents 4 3x = 8 x+1 (2 2 ) 3x = (2 3 ) x+1 rewrite w/ same base 2 6x = 2 3x+3 6x = 3x+3 x = 1 Check → 4 3*1 = = 64

Your turn! 2 4x = 32 x-1 2 4x = (2 5 ) x-1 4x = 5x-5 5 = x Be sure to check your answer!!!

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

4 x = 15 log 4 4 x = log 4 15 x = log 4 15 = log15/log4 ≈ 1.95 Using LogUsing LN

10 2x-3 +4 = x-3 = 17 log x-3 = log x-3 = log 17 2x = 3 + log17 x = ½(3 + log17) ≈ 2.115

Solving an Exponential Equation containing e 40e 0.6x -3= 237

Solving Log Equations To solve use the property for logs w/ the same base use the one-to one property: If log b x = log b y, then x = y

log 3 (5x-1) = log 3 (x+7) 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check log 3 (5*2-1) = log 3 (2+7) log 3 9 = log 3 9

log 5 (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

log5x + log(x+1)=2 log (5x)(x+1) = 2 (product property) log (5x 2 – 5x) = 2 5x 2 - 5x = 100 x 2 – 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