Supplement for M122 College Algebra 

Exponential and Logarithmic Equations and Applications 

 

One-to-one Properties: 

 

If MNbb=, then ______ = _______. If loglogbMN=, then ______ = _______. 

 

Solving Exponential and Logarithmic Equations using the one-to-one properties. 

 

1. Solve: 





 

 

 

 

 

 

 

2. Solve: 
421x-=

 

 

 

 

 

 

 

3. Solve: 
35128x+= 

 

 

 

 

 

 

 

 

 

4. Solve 2loglog16x=

 

 

 

 

 

 

Rewrite 125 as 5 raised to a power. 

 

Use 1-1 property. 

Rewrite 16 as 2 raised to a power. 

 

Use 1-1 property 

 

Solve for x. 

Rewrite 18 
as 2 raised to a power. 

 

Use 1-1 property. 

 

Solve for x. 

Use the power property of 
logarithms. 

 

Use 1-1 property. 

 

Solve for x. 

512555xxx=
=
=

 

 

 Why can x not equal ? 4-

 

 

 Verify your solution graphically: 

 

 

 

 

Solving Exponential and Logarithmic Equations using other properties: 

 

 

1. Solve: 441log(3)log(3)2xx++-= 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


2. Solve: 542xe+=


 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use the sum property to combine into 
a single logarithm. 

 

Change to exponential form. 

 

Solve the quadratic equation for x. 

 

Check the domain. 

Solve for ex. 

 

Take the ln of both sides. 

 

Use inverse property. 



3. Solve: 121616log(3)log(2)xx+--= 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 




4. Solve: 
245xx-= 

 

 

 

 

 

 

 

 

 

 

 

 

 

More Practice with Exponential and Logarithmic Equations: 

 

1. 2. ln48xe=(3)5x+= 3. l og(2)52x-+=

 

 

 

 

 

4. 5. ln35x+=424xe-+= 6. lo( 4

 

 

 

Use the difference property to 
combine into a single logarithm. 

 

Change to exponential form. 

 

Solve the equation for x. 

 

Check the domain. 

Take the ln of both sides. 

 

Use the power property. 

 

Solve for x. 


Logarithmic and Exponential Applications 

 

1. A colony of bacteria increases exponentially. The initial quantity of bacteria is 25 grams and 
the number of bacteria doubles every 5 hours. 


 

a) When will the quantity equal 50 grams? (You can answer this without using a formula) 


 

 

 

b) When will the quantity equal 100 grams? (You can answer this without using a 
formula) 


 

 

 

c) When will the quantity equal 75 grams? (You will need to do some work to solve this 
one) 


 

Step 1: Determine the rate of growth, k. 

 

 

 

 

 

 

 

 

 

 

 

 

Step 2: Determine the time needed to grow to 75 grams. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


2. Suppose a material has a half-life of 1000 years. This means that every 1000 years, half of the 
material will decay. How long will it take 100 grams to decay to 5 grams? 

 

Step 1: Find the rate of decay (k) 

 

 

 

 

 

 

 

 

Step 2: Find the time needed for 100 grams to decay to 5 grams. 

 

 

 

 

 

 

 

 

 

 

 

 

3. Suppose the initial quantity of a material is 80 grams and decays to 60 grams in 200 years. Find 
the half-life of the material. 

 

Step 1: Find the rate of decay (k) 

 

 

 

 

 

 

 

 

Step 2: Find the time needed for half the material to decay. 

 

 

 

 

 

 

 

 

 

 


4. A company’s revenue for a new product after x months is ()2ln(2)Rxx=+. The cost of 
development and production after x months is ()ln(652)Cxx=+. Revenue and Cost are 
both in millions of dollars. 

 

a) When will the company break even? Solve algebraically. 


 

 

 

 

 

 

 

 

 

 

 

b) Write the profit function ()()()PxRxCx=- and simplify using properties of logarithms. 


 

 

 

 

 

 

 


c) When will the profit equal $2 million? Solve algebraically or graphically. 


 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 





5. The number of students (in thousands) attending school A ( x years after 2000 ) is given by 
. The number attending school B is given by . 5()log(43)Axx=5()log(89)Bxx=- 



a) Write a function which gives the total number of students attending both schools 
and simplify. 
()Tx 


 

 

 

 


b) When will total attendance for both schools be 6 thousand students? Solve 
algebraically. 


 


 

 

 

 

 

 

 

 

 

 


c) What will be the total number of students attending both schools in 2006?