Logarithm function and Exponential functions

Logarithm function
The logarithm function with base is the function where and
The function is defined for all .
Here is its graph for any base b.

Note the following:
1. For any base, the x-intercept is 1. ?
2. The graph passes through the point ?
3. The graph is below the x-axis -- the logarithm is negative -- for
4. The function is defined only for positive values of x.
5. The range of the function is all real numbers.
6. The negative y-axis is a vertical asymptote.
7.



11. For each strictly positive real number and , different from 1, we have

The natural logarithm
The system of natural logarithms has the number called as its base. e is an irrational number; its decimal value is approximately 2.71828182845904.
To indicate the natural logarithm of a number we write " ." means
So we have





Derivative of natural logarithm function
If is a function , then

Example 1: Find derivatives of the functions








Exponential functions
For any positive number , there is a function called an exponential function that is defined as
For example



Now, let’s talk about some of the properties of exponential functions.
1. The graph of will always contain the point . Or put another way, regardless of the value of .
2. For every possible . Note that this implies that .
3. If then the graph of will decrease as we move from left to right.
4. If then the graph of will increase as we move from left to right.
5. If then
Basic rules for exponents



Natural exponential function
The function is often called exponential function or natural exponential function which is an important function. The exponential function is the inverse of the logarithm function
Derivatives of exponential function
If is a function , then

Example 2: Find of the functions















Solving exponential and logarithm equations
Logarithms are the "opposite" of exponentials. In practical terms, I have found it useful to think of logarithm in terms of the relationship:


Physical application:
i- Radioactive Decay: The amount of a radioactive element at time is given by:

Where is the initial amount of the element and is the constant of proportionality.
Example 3: The radioactive element radium-226 has a half-life of 1620 years. If a
sample initially contains 120 gm find the constant





Example 4: The radioactive element Iodine-131 has a half-life of 8 days. If a sample
initially contains 5 gm . Find a function which gives the amount at any time





ii- Newton’s low of cooling
The temperature of an object at time is given by:
Where the temperature of the surrounding medium and are constants.
Example 5: Placed a metal bar, at a temperature of 100°F in a room with constant temperature of 0°F. After 20 minutes the temperature of the bar is 50°F. Find
Solution:



To find


Example 6: A body at an unknown temperature is placed in a room which is held at a constant temperature of 30° F. If after 10 min the temperature of the body is 0° F and after
20 min the temperature of the body is 15° F, find the unknown initial temperature.
Solution:













Find derivative in each of the following problems




5. The radioactive element Chromium- 51, has a half-life of 27.7 days. If a sample initially contains 75 milligrams. Find a function which gives the amount at any time
6. A 40°F roast is cooked in a 350°F oven. After 2 hours, the temperature of the roast is
125°F.
a. Find a formula for the temperature of the roast as a function of .
b. The roast is done when the internal temperature reaches 165°F. When will the roast be done?
7. A body at a temperature of 50° F is placed in an oven whose temperature is kept at 150° F. If after 10 minutes the temperature of the body is 75° F, find the time required for the body to reach a temperature of 100° F.