Exponential and Log Equations

I. Homework

II.  Equations that Involve Logs

Step by Step Method

Step 1:  Contract to a single log

Step 2: Get the log by itself

Step 3:  Exponentiate both sides with the appropriate base

Step 4:  Solve

Step 5:  Check your solution for domain errors

Example: Solve

log₅ x + log₅ (x + 2) = log₅ (x + 6)

1)   log₅ x + log₅ (x + 2) - log₅ (x + 6) = 0

log₅ x (x + 2) - log₅ (x + 6) = 0

log₅ x (x + 2)/(x + 6) = 0

2)  Already done.

3)  x(x + 2)/(x + 6) = 5⁰ = 1

4)   x(x + 2) = x + 6

x² + 2x - x - 6 = 0

 x² + x - 6 = 0

(x - 2)(x + 3) = 0

x = 2 or x = -3

5)  Note that -3 is not in the domain of the first log hence the only solution is x = 2

Exercises:  Solve

A)  log(x + 6) + 1 = 2log(3x - 2)

B)  1/2 log(x + 3) + log2 = 1

III.  Exponential Equations

Step 1:  Isolate the exponential

Step 2:  Take the appropriate log of both sides.

Step 3:  Solve

Example: Solve

 4e^-7x = 15 

1)  e^-7x = 15

2)  lne^-7x = ln15

3)  -7x = ln15

4)  x = (ln15)/-7

Exercises:  Solve

A)  1 + 2e^x = 9

B)  (10^{x - 4})/e^2x - 4 = 0

C)  (lnx)²  = ln(x²)

D)  2^3x + 4(2^-3x) = 5