Solving Exponential Equations using Logarithms

MHF4UIFriday October 19th , 2012

We can use Logs to Solve Equations!

Logarithms are particularly useful in solving equations with unknown exponents. As a rule, you can take the log of both sides

of an equation.

Examples:

What if the equation gives multiple solutions?

You will find many cases where you solve an exponential equation that produces two or more distinct solutions.

Therefore x=3 or x=2 are both possible solutions to this equation

Watch out for erroneous solutions!

The important thing to watch out for when equations give multiple roots, are the cases where values of x will be less than zero. These

solutions will be inadmissible.We must remember that is only defined for x>0.

However since we cannot have a negative x value in our logarithmic function, we must declare x = -5 to be an inadmissible

solution.Therefore x = 5

You must declare all erroneous/inadmissible solutions.

Let’s try some trickier examples

Be sure to state all inadmissible/extraneous roots!

Let’s try some trickier examples

Be sure to state all inadmissible/extraneous roots!

We can use Logarithms to Solve Exponential Functions

Taking the logarithm of a function will be particularly useful when we are solving problems of Exponential Growth and Decay.

Problems involving Exponential GrowthPopulation Growth, Bacterial Cultures, Financial problems (RRSPs, Savings)

Problems involving Exponential DecayValue of a vehicle (Depreciation), Half-Life of Radioactive Chemicals, Carbon Dating

In these types of problems we will be analyzing when quantities increase or decrease in the future proportional to the quantity in the present.

Exponential Functions make our Lives Easier!

Consider the population of deer found in Algonquin Park. It is estimated that the current population is 15,000 deer and based on

past rates of growth the population is expanding at 3% per year.

2013 15,000(1.03) = 15,450 15,000(1.032014 15,450(1.03) = 15,914 15,000(1.032015 15,914(1.03) = 16,391 15,000(1.03

We can express this data using the following function:Population at n-years = 15,000(1 + 0.03

The General Formula for Problems of Exponential Growth and Decay

The standard form of an exponential growth or decay is:

Where, is the initial amount at t = 0b is the factor of growth (1 + i) or decay (1 – i)i is the percent rate of growth or decay (must be expressed as a decimal)t is the number of growth or decay periodsA(t) is the amount at time t

World Population Example

The world population doubles every 11 years. The population was 7 billion in 2013.

What will the Population be in 2023?

When will the Population be 25 Billion?

At what rate would the Population need to grow to reach 50 Billion by the year 2033?

World Population Example (Solution)

What will the Population be in 2023? = b = t = A(t) =

World Population Example (Solution)

When will the Population be 25 Billion? = b = t = A(t) =

World Population Example (Solution)

At what rate would the Population need to grow to reach 50 Billion by the year 2033?

= b = t = A(t) =

Car Depreciation Example

Mr. Richmire just bought a brand new Porsche 911 Carrera for $93,000. It is estimate that the car depreciates by 15% per year.

What will the value of the car be in 10 years?

When will the value of the car reach $20,000?

Car Depreciation (Solution)

What will the value of the car be in 10 years? = b = t = A(t) =

Car Depreciation (Solution)

When will the value of the car reach $20,000? = b = t = A(t) =

Half-Life Example

An isotope of radium has a half-life of 1620 years. You currently have 10mg.

How much radium will be remaining after 50 years? 500 years?

A 10mg sample of uranium decays to 8mg in 4 minutes, determine the half-life of uranium.

Half-Life (Solution)

How much radium will be remaining after 50 years? 500 years? = b = t = A(t) =

Half-Life (Solution)

A 10mg sample of uranium decays to 8mg in 4 minutes, determine the half-life of uranium.

= b = t = A(t) =

Homework Questions:

• Textbook Chapter 7.2• Part A: 2, 3, 4, 6, 7• Part B: 8, 9, 13, 15, 16a