7.6 - The Natural Base, e

7.6 - The Natural Base, e

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Objective

To evaluate and simplify natural logarithmic expressions

To solve equations using natural logarithms

Essential Understanding

Example

Example

Mathematicians have been familiar with the number e for centuries.

It has applications in finance and science, and is the base of the natural log, known as ln.

It is the limit as x gets very large of  $y=\left(1+\frac{1}{x}\right)^x$  

The graph of  $y=e^x$  

Has all the same properties of logs with any other base that you have learned about in this unit. 

Fill in the Blank

Condense to a single log: $$\ln3+\ln16$$

Multiple Choice

Expand the log: $$\ln\left(\frac{a^3}{b}\right)^2$$

• $$\ln a^3+\ln b^2$$

• $$6\ln a-2\ln b$$

• $$\ln a^6-\ln b^2$$

• $$\ln2a^3-\ln b$$

The graph of  $y=\ln\ x$  

Economics application

Example 1

You try

Open Ended

What is the value of an investment of $100 invested at 3% interest for 8 years and compounded continuously?

Did you get $127.12?

One thing to watch for is make sure you convert the interest rate to a decimal. So 3% interest goes into the formula as 0.03.

The natural log also has scientific applications

• The half-life of a substance is the amount of time it takes for half of the substance to convert to another substance during the process of decay.

Natural decay is modeled by this formula.

Exponential growth function

Example

Fill in the Blank

The amount of bacteria in a culture can be modeled by $$f\left(t\right)=400e^{0.15t}$$ , where t is th enumber of hours since 7:00 am. Find the bacteria count at 3:30 pm on the same day. Round to the nearest whole number

Poll

Select the basketball footwear that most closely resembles how you feel about being able to solve equations and word problems involving the natural log: