Powers and Logarithms

Using Logarithms to Compute Powers

Logarithms are useful tools for calculating powers and roots, especially before calculators were common. They allow us to transform multiplication and exponentiation into addition and multiplication, respectively.

• Definition: The logarithm base 10 of a number x, written as log(x), is the exponent to which 10 must be raised to obtain x.

• Example: To find using logs, recall that . Using logarithms:

Find from a table or calculator, divide by 2, then find the antilog to get the answer.

Binary Exponents and Logarithms

Computers use base 2 (binary) rather than base 10. Knowing that helps estimate powers of 2.

• Key Fact:

• Example: To compute :

So

• Memory Sizes:

• Population Example: If a population doubles every generation, after 20 generations it multiplies by .

• Quick Estimation: is approximately 1 followed by zeros.

• Example: : , so followed by 18 zeros ().

Solving Exponential Equations with Logarithms

General Method

When the variable is in the exponent, logarithms allow us to solve for it.

1. Isolate the exponential term.

2. Take the logarithm of both sides.

3. Use the power rule: to bring the exponent down.

4. Solve for the variable.

• Example 1: Solve

Take log of both sides: Since ,

• Example 2: Solve

• Example 3: Solve

First, isolate the exponential term: Take logs:

• Example 4: Solve

This equation is more complex and may require iterative or graphical methods if appears on both sides. If is only in the exponent, isolate and proceed as above.

Compound Interest

Exponential Growth in Finance

Compound interest is a classic application of exponential functions. The amount after years, with principal and annual interest rate (as a decimal), compounded once per year, is:

• Example 1: , ,

• Example 2: , ,

• Finding Time to Reach a Target: To find how long it takes for an investment to grow from to at rate :

Take logs:

• Example: , ,

years (dont understand)

Applications: Population Growth

Exponential Growth in Biology

Many populations grow exponentially, meaning each generation multiplies the previous population by a constant factor.

• General Formula: If the initial population is and each generation multiplies by , after generations:

• Example 1: 100 bunnies, each generation triples ():

• Finding Number of Generations to Reach a Target:

Set target, solve for : Take logs:

• Example: How many generations until there are 107 bunnies?

generations (which suggests a check of the numbers; likely the target should be much larger for a meaningful answer).

Applications: Epidemic Growth

Exponential Growth in Disease Spread

In epidemiology, the number of infected individuals often grows exponentially in the early stages of an outbreak.

• Example: 5 infected cows, number doubles every 3 days. How many days until there are 80 infected cows?

Let be the number of 3-day periods. (since ) Total days = days

Additional info: These examples illustrate the use of logarithms and exponents in solving real-world problems, including finance, population biology, and epidemiology. Mastery of these techniques is foundational for calculus students, especially in understanding exponential growth and decay, and in preparation for differential equations.