1. Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5
Logarithmic, Exponential, and Other Transcendental Functions

2. Chapter 5 Section 5: Bases other than e and Applications
MA.C Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.

3. Base a
If a is a positive real number (𝑎≠1) and 𝑥 is any real number, then the exponential function to the base 𝒂 is denoted 𝑎𝑥 and is defined by
𝑎𝑥 = 𝑒 ln 𝑎 𝑥
If 𝑎 = 1, then 𝑦= 1 𝑥 =1 is a constant function.
The usual properties still apply
𝑎 0 =1 𝑎 𝑥 𝑎 𝑦 = 𝑎 𝑥+𝑦
𝑎 𝑥 𝑎 𝑦 = 𝑎 𝑥−𝑦 ( 𝑎 𝑥 ) 𝑦 = 𝑎 𝑥𝑦

4. log 𝑎 𝑥 = 1 ln 𝑎 ∗ ln 𝑥 = ln 𝑥 ln 𝑎
Log base a
If a is a positive real number (𝑎≠1) and 𝑥 is any positive real number, then the logarithmic function to the base 𝒂 is denoted by log 𝑎 𝑥 and is defined as
log 𝑎 𝑥 = 1 ln 𝑎 ∗ ln 𝑥 = ln 𝑥 ln 𝑎
The following similar properties apply
log 𝑎 1 =0 log 𝑎 𝑥𝑦 = log 𝑎 𝑥 + log 𝑎 𝑦
log 𝑎 𝑥 𝑛 =𝑛 log 𝑎 𝑥 log 𝑎 𝑥 𝑦 = log 𝑎 𝑥 − log 𝑎 𝑦

5. Inverse Functions Try: log 5 25 =𝑥 log 2 1 64 =𝑥 3 𝑥 = 1 81
𝑦= 𝑎 𝑥 if and only if 𝑥= log 𝑎 𝑦
𝑎 log 𝑎 𝑥 =𝑥, for 𝑥>0
log 𝑎 𝑎 𝑥 =𝑥, for all 𝑥
Examples:
3 𝑥 = 1 81
log 𝑥 = log
𝑥=−4
log 2 𝑥 = −4
𝑥= 2 −4
𝑥= 1 16
1 81 = 3 −4

6. Other Bases Bases are used to solve: Earthquake measurement Half-life
Depreciation
Interest Rates
And many more

7. Half Life
When modeling the half-life of a radioactive element, it is convenient to use ½ as the base of the exponential model.
If t = time and d = half-life of sample, then 𝑡/𝑑 is the model of decay.
For example: Gold-198 has a half-life of days

8. Depreciation Depreciation works in the same way.
It is estimated that a computer depreciates to one half its purchase price in one year.
Thus the model is (½)t/1. In 3 years, the computer will be work (½)3 or 1/8 of its original cost.
Problem: The time in which a machine depreciates to one-half its purchase price is 2 years. Find a model which yields the fraction of the purchase price as a function of time and determine that fraction after 5 years.

9. Annual Interest Rates 𝐴=𝑃 𝑒 𝑟𝑡 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
Suppose 𝑃 dollars are invested in an account at an annual interest rate of 𝑟 (in decimal form) compounded 𝑛 times a year for 𝑡 years, then
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
The formula for continuous compounding is
𝐴=𝑃 𝑒 𝑟𝑡

10. Limit Involving e
The continuously compounding formula is derived using the following theorem: lim 𝑥→∞ 𝑥 𝑥 = lim 𝑥→∞ 𝑥+1 𝑥 𝑥 =𝑒

11. Comparing Rates
The government decides to give $1000 to students for graduating from high school. You decide to save the money for four years in case you cannot find a job immediately. The Bank of Newton is offering the following savings account options. In which account should you invest your money?
Savings account at 5.21% compounded annually
CD at 5.12% compounded continuously
Money Market at 5.13% compounded daily
Checking account at 4.98% compounded monthly
Savings $1106.9
CD $
Money Market $
Checking $

12. Differentiation and Integration
(1) You can use the definitions of ax and loga x as well as the rules already learned for natural log and exponential functions to differentiate,
(2) use logarithmic differentiation
(3) use the differentiation rules for bases other than e
Let 𝑎 be a positive real number (𝑎≠1) and let 𝑢 be a differentiable function of 𝑥.
𝑑 𝑑𝑥 𝑎 𝑥 = ln 𝑎 𝑎 𝑥
𝑑 𝑑𝑥 𝑎 𝑢 = ln 𝑎 𝑎 𝑢 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 log 𝑎 𝑥 = 1 ln 𝑎 𝑥
𝑑 𝑑𝑥 log 𝑎 𝑢 = 1 ln 𝑎 𝑢 ∙ 𝑑𝑢 𝑑𝑥

13. Differentiation and Integration
Find the derivative of:
y = 23x
y = log5 (sin x)
y = x(7-3x)
Let 𝑎 be a positive real number (𝑎≠1) and let 𝑢 be a differentiable function of 𝑥.
𝑑 𝑑𝑥 𝑎 𝑥 = ln 𝑎 𝑎 𝑥
𝑑 𝑑𝑥 𝑎 𝑢 = ln 𝑎 𝑎 𝑢 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 log 𝑎 𝑥 = 1 ln 𝑎 𝑥
𝑑 𝑑𝑥 log 𝑎 𝑢 = 1 ln 𝑎 𝑢 ∙ 𝑑𝑢 𝑑𝑥
Tractor example: Derivative is the rate of change of the cost. Integral tells you how much money you have “lost” in a time frame.

14. Differentiation and Integration
To integrate, you can first use the equation 𝑎 𝑥 = 𝑒 ln 𝑎 𝑥 or you can use the integration formula: 𝑎 𝑥 𝑑𝑥 = 1 ln 𝑎 𝑎 𝑥 +𝐶
Examples:
Find 𝑥 𝑑𝑥
Find −𝑥 𝑑𝑥

15. Comparing Constants and Variables
When taking derivatives, pay close attention to the arrangement of the function.
Typically, if you have 𝑥 raised to a function of 𝑥, you should use logarithmic differentiation.
e^e  0 (no x in problem)
e^x  e^x
x^e  e*x^(e-1)
x^x = y  ln x^x = ln y  x*ln x = ln y (x)(1/x) + (ln x)(1) = (1/y) y’  1+ln x = (1/y) y’  (x^x)(1+ln x)

16. Homework Page 368-370 3-33 every three, 97, 101