Applications of 'e' and 'ln'

7.6 - The Natural Base, e

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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 elarned about in this unit. 

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

If k = 0.1536, and the population continued to grow at this rate, what would be the population in 2010?

Multiple Choice

Condense the expression as a single logarithm.

 $\ln4+\ln3x$  

•  $\ln7x$  

•  $\ln\left(3x\right)^4$  

•  $\ln12x$  

Multiple Choice

Write the equation in logarithmic form.
 $e^3=x$  

•  $\ln x=3$  

•  $\ln3=x$  

Multiple Choice

You deposit $5,000 in an account that earns 3% interest compounded continuously. Which equation could be used to determine how much money would be in the account after 8 years?

• $A=5000e^{3\cdot8}$

• $A=5000e\cdot3\cdot8$

• $A=5000e\cdot0.03\cdot8$

• $A=5000e^{.03\cdot8}$

Multiple Choice

Simplify the expression, leaving e in the answer.

 $e^3\cdot e^5$  

•  $e^{15}$  

•  $e^8$  

•  $2e^8$  

•  $2e^{15}$  

Multiple Choice

Rewrite in exponential form:

ln(2) = x

• 2^x = 10

• 10² = x

• e^x = 2

• e² = x

Multiple Choice

Solve the following equation for x. Round your solution to two decimal places.

ln2x - ln41 = 2

• x = 0.36

• x = 151.48

• x = 41

• None of these are solutions