8-5 Exponential & Logarithmic Equations Strategies and Practice

Objectives – Use like bases to solve exponential equations. – Use logarithms to solve exponential equations. – Use the definition of a logarithm to solve logarithmic equations. – Use the one-to-one property of logarithms to solve logarithmic equations.

Use like bases to solve exponential equations Equal bases must have equal exponents EX: Given 3 x-1 = 3 2x + 1

 If possible, rewrite to make bases equal EX: Given 2 -x = 4 x+1

Isolate function if needed— 3(2 x )=48

You try… 1. 4 x = x-2 = 25 x 3. 6(3 x+1 ) = e –x 2 = e -3x - 4

Solving Logarithmic Equations Convert to exponential (inverse) form EX: Solve: 2log 5 3x = 4

Now you try…. log x = 6 log 5x = 3

Solving Logarithmic Equations Use Properties of Logs to condense EX: Solve: log 4 x + log 4 (x-1) = ½

You try… Solve lnx+ln(x-3) = 1 Solve log x + log (x + 2) = log (x + 6)

Double-Sided Log Equations Equate powers (domain solutions only) EX: Solve: log 5 (5x-1) = log 5 (x+7) EX: Solve: ln(x-2) + ln(2x-3) = 2lnx

You try… 1. Solve ln3x 2 = lnx 2. Solve log 6 (3x + 14) – log 6 5 = log 6 2x 3. Solve log 2 x+log 2 (x+5) = log 2 (x+4)

Exponentials of Unequal Bases Use logarithm (inverse function) of same base on both sides of equation EX: Solve: e x = 72

Now you try…. 2e x + 2 = 12 e x – 9 = e x = 5

EX: Solve: 7 x-1 = 12

You try… 1. Solve 3(2 x ) = Solve 3 2t-5 = Solve e 2x = 5 4. Solve e x + 5 = 60

Solving Logarithmic Equations Convert to exponential (inverse) form EX: Solve: lnx = -1/2

Now you try ln (2x – 1) = 0 ln x = -3 3ln 5x = 10 Solve lnx+ln(x-3) = 1

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!