Lesson 14- Logarithm Equations and Identities

Today's lesson had two methods for solving equations, log= # and log=log

Method 1: Log= #                                                                                                                                             

log2X = 3         1)  put into B^e = A form

2^3 = x

x= 8 

example2:

log3(x-4) = 2 + log3X

log3(x-4) = 2               1) put into B^e = A form

         x

3^2 = x-4              2) multiply both sides by x

            x

x(3^2) = (x-4/x)x       3) simplify

9x = x-4

8x= -4

x= -1/2  -- reject       

x= undefined 

*note: negative answers are tricky, they must become positive in original equation or are rejected.

       

Method 2: log = log                                                                                                                       

log5(x-3) + log5x = log5 10     1) create only one log

log5(x-3)x = log5 10            2) Drop logs

(x-3)x = 10

x^2 - 3x = 10

x^2 -3x -10 = 0

(x-5)(x+2)

x= 5, -2 -- reject ( negative does not work in eq'n.)

x=5

Identities:

Equation        vs.         Identity

2x+3 = 5                    x + x = 2x

x=1                            x = all real numbers

x= specific #                rejects allowed

Prove the identity and state the value(s) of x for which it is true:   ( make both sides equal)

  

logx + log(x +3) = log(x^2 +3x)

    LEFT SIDE           RIGHT SIDE

Left side:  logx(x+3)

               logx^2 + 3x = Right side

x must be great the 0     x>0