Home Exponents Exponential Equations I Exponential Functions The Exponential Function Base e The Logarithm Functions Logarithmic & Exponential Equations II Applications:Growth & Decay Review&Test Review&Test Review1 Review2

The following properties of logarithms are important and used frequently in our study of logarithms.  They correspond closely to

our rules for exponents studied earlier.

 Exponential Form Logarithmic Form 34 = 81 log381 = 4 52 = 25 log525 = 2 25 = 32 log232 = 5 103 = 1000 log101000 = 3

UNIT 4  : EXPONENTIAL & LOGARITHMIC FUNCTIONS REVIEW QUESTIONS

Solutions

 Interval Test Number - 4 - 1 4 Sign of (x – 3 ) ( - ) ( - ) ( + ) Sign of (x + 3) ( - ) ( + ) ( + ) Sign of (x – 3 )(x + 3) ( + ) ( - ) ( + )

 Interval Test Number - 3 - 1 5 Sign of (x + 2) ( - ) ( + ) ( + ) Sign of (x – 3) ( - ) ( - ) ( + ) Sign of (x + 2)(x – 3) ( + ) ( - ) ( + )

 Interval Test Number - 5 0 3 Sign of (2x – 5 ) ( - ) ( - ) ( + ) Sign of (x + 4) ( - ) ( + ) ( + ) Sign of (2x – 5)/(x + 4) ( + ) ( - ) ( + )

11.  Solve for x.

Solutions:

12.  Solve for x.

Solutions:

13.  Solve for x.

Solutions:

14.  Solve for x.

Solutions:

15.  A biologist grows a colony of bacteria in a Petri dish.  Under ideal conditions, the doubling period is 3 h.  How long will it take for the colony to grow to 32 times its original size?

Recall:

 A  = ? A0 = ?    t = ?   d = 3 h

,  where

·         A is the number of bacteria after the given time frame

·         A0 is the starting number of bacteria

·         2 is the growth factor

·         t is the total time elapsed in the experiment

·         d is the doubling period

Solution: In this problem, we are asked to find the time it takes to grow to 32 times its original size.  We are not told the exact starting and final numbers of bacteria.  The key here is that whatever the starting amount is, the final amount will be exactly 32 times larger.  Therefore we let the starting amount be 1 g .  Hence the final amount will be 32 g.

We now have the following information in the box at right.

 A  = 32 g A0 = 1 g    t = ?   d = 3 h

Therefore it will take 15 h to grow to 32 times its original size.

16. The half-life of radium is 1600 years.  What fraction of radium remains from a sample after 12 800 years?

Solution: Again let the starting mass be 1 g.

 A  = ? A0 = 1 g    t = 12 800 years   h = 1 600 years

Recall:

Therefore   of the original amount remains after 12 800 years.

 A  = 3.125 g A0 = 100g    t = 2 000 years   h = ?