Section+3.5

3.5 Equation Solving and Modeling

One-to-One Properties you should know: Example. Solve: 20× (1/2)^(x/3)=5
 * For any exponential function f(x)= b^x
 * If b^u= b^v, then u=v
 * For any logarithmic function f(x)= logbV
 * If logbU=logbV, then U=V

1/2^(x/3) = 1/4

1/2^(x/3)= (1/2)^2

x/3= 2

x=6

Example 2. Solve: ln(3x-2)(x-1)= 2lnx

ln [(3x-2)(x-1)]= lnx^2

(3x-2)(x-1)= x^2

3x^ -3x-2x+2= x^2

2x^2-5x+2=0

(2x-1)(x-2)=0

x= 1/2, or 2 x=1/2 proves to be an extraneous solution

Generally: Example. 3^x= 10
 * 1) Isolate the power or the log
 * 2) Use the one-to-one property or convert the equation to its equivalent form
 * i. e. log to exponent or exponent to log

log(3)10=x

(log10)/(log3)=x

x= 2.096

Logarithmic Re-Expression

1. logarithmic equation- y= alnx+b 2. Exponential equation- y= a×b^x 3. Power function equation- y= kx^a

Change the ordered pairs (x,y) into: Whichever set of new ordered pair gives the best linear equation, use the corresponding function for the original
 * 1) (lnx, y) Logarithmic
 * (x, lny) Exponential
 * 1) (lnx, lny) Power

Proof: y= ax+b → y= alnx+b, logarithmic

y=ax+b → lny= ax+b, (solve for y) exponential

y= ax+b → lny= alnx+b, (solve for y) power