Exponents 

Base^exponent= Answer

the base and exponent can be any real number

D:   all real numbers

the answer must be positive

R:   y: y > 0   

Exponent Rules 

If the bases are the same: 

To multiply, add the exponents

(x⁴)(x⁵) = x⁹        

To divide, subtract the exponents

Power of a power, multiply the exponents 

(a⁷)³ = a²¹           

Logarithms 

log_baseAnswer = Exponent

log_baseargument= answer

the base and argument have to be positive

D:   x: x > 0

the answer can be any real number

R:   all real numbers  

Log Rules 

If the bases are the same: 

Multiplying can be split into adding 

log_bMN = log_bM + log_bN 

Adding logs of the same base can be combined into multiplying the arguments 

log_bM + log_bN = log_bMN   

Dividing can be split into subtracting 

log_b(M/N) = log_bM - log_bN 

Subtracting with the same base can be combined into dividing the arguments

log_bM - log_bN = log_b(M/N)    

Power of a log can be changed to multiplying 

log_bM^k = k log_bM 

Multiplying a number times a log can be changed to a power of the log

log k log_bM = log_bM^k    

Strategies for Solving Log Problems 

1)  Change from log notation to exponential notation. 

Example:          log₂₇j=2/3      log_baseanswer=exponent

                        27^2/3=j

                   9=j 

Example:        ln x² = - 2           log_base=eanswer= exponent                       

                       e^-2 = x²                To get x by itself, take the square root of both sides.    

                                                          Since x is squared, there will be two answers, and you need to +/- the left side                  

                  (e^-2)^1/2 = (x²)^1/2      or  - (e^-2)^1/2 = (x²)^1/2                        

                  e^-1 = x                  or – e^-1 = x                  

                  x= 1/e or -1/e 

2)  Use inverse operations

Example:        log₆6⁷         6 to what power gives you 6 to the 7^th power?                  

                   7 

Example:        3^log₃⁸          a log in the exponent with the same base undo each other                  

                    8 

3)  Take the common log of both sides.

Example:        5^3x= 786

log 5^3x=  log 786       a power of a log can be changed to multiplying

3x(log 5)= log 786   divide both sides by log 5 & 3

x= (log 786)              plug it into your calculator   

     (log 5)  3        

x= 1.380804387       round to whatever the directions say  

4)  Use the log properties to simplify and solve.

                   8=x² – x – 12   move the 8 over and get a 0 on one side                  

                  -8              -8                  

                   0= x² – x – 20   factor                  

                   0=(x – 5)(x + 4)   use the zero product rule                  

                   x – 5 = 0          x + 4 = 0 get x by itself in each equation        

                   x=5   x= -4 check to make sure the original arguments are positive              

                  x=5             negative 4 does not work in either original argument  

5)  Use the change of base formula:  log_ax= log_bx    

                                                                  log_ba   

Example:       3^x= 21          change from exponential notation to log notation                    

                  log₃21=x      change base to either 10 or e                   

                  log 21  = x    new base is 10, plug into calculator                       

                  log 3                   

                  2.7712= x

Example:       log₅16        argument of numerator becomes new argument

                       log₅4         argument of denominator become new base

                  log₄16        4 to what power makes 16?

Looking for a tutor in Santa Cruz,

Capitola, or Scotts Valley, California?  

Please check my "current openings" page to see if I am accepting new students.