Lesson 3
Unit e Ln 2: Exponential vs. Logarithmic expressions
• A logarithmic function is the inverse of an exponential function.
• It is used to find the exponent of a logarithmic function
Definition of Logarithmic Function:
For x > 0, a > 0, and a ≠1, [pic] if and only if,[pic].
[pic] because [pic]
The function [pic] is called the logarithmic function with base a.
(Read as “log base a of x”)
CONVERTING BETWEEN LOGS AND EXPONENTIAL EXPRESSIONS:
Convert the following functions into either exponential or logarithmic form:
|Exponential equation |Logarithmic equation |
|35=243 |log3243=5 |
|10-1 = 0.1 | |
| |log64(4)=1/3 |
|45 = 1024 | |
| |log8512=3 |
|65 = 7776 | |
Evaluating Logs with the Definition of Logarithm:
Logarithm with base 10 such as [pic] is called common logarithm and is written as [pic]
(base 10 is “invisible”) Common Logarithm can be evaluated on the calculator.
**NOTE: LOG key on your calculator indicates common log with base 10 only!**
Examples: Evaluate the following functions using the definition of logs.
1. [pic] 2. [pic] 3. [pic] 4. [pic]
2f(x) = 32
25 = 32
f(x) = 5
Examples: Evaluate the following expressions, using the definition of logs:
5. log5(25). 6. log2(8). 7. log64(4). 8. log6(6). 9. log4(–16). 10. log2(0).
5y=25
52 = 25
y=2
Important Logarithmic Identities are:
[pic] because [pic]
[pic] because [pic]
[pic] because [pic]
[pic] [pic] **there is no way we can raise a positive base to some power and end up with zero**
[pic] [pic] ( x = y
Evaluating Logs that do not simplify easily
There is a formula that you can use with logarithms that do not simplify easily. For instance, try to evaluate[pic].
Change of Base Formula: For any positive real numbers [pic]
This formula puts both logs into the base of 10, which your calculator can help you evaluate.
[pic] = log108 / log103
This can be evaluated using the log button on your calculator (because it’s base 10!)
log38 = 0.90/0.48 = 1.89
Examples: Evaluate the following expressions, using the change of base formula.
a. log49 b. [pic] c. [pic]
log(9)/log(4) log(10) / log(12) log(15)/log(9)
1.58 0.93 1.23
Where do we use logarithmic functions?
Richter Scale – The magnitude of an earthquake is [pic] where I is intensity of the earthquake and S is intensity of a standard earthquake
Decibels (dB) "Loudness" is measured in decibels. The formula for the loudness of a sound is given by
"dB = 10log[ I ÷ I0 ]" where I0 is the intensity of "threshold sound", or sound that can barely be perceived.
The pH scale - Chemists define the acidity or alkalinity of a substance according to the formula
"pH = –log[H+]" where [H+] is the hydrogen ion concentration, measured in moles per liter.
Mixed Practice: Evaluate the following expressions/equations, using the definitions and properties of logs.
1. [pic] 2. [pic] 3. [pic] 4. [pic]
=14 =x no solution -0.6
5. [pic] 6. [pic] 7. [pic] 8. [pic]
y=-2 x=16 x=1.26 y=-3
9. [pic] 10. [pic]=x 11. [pic] 12. [pic]
X=9 x=-2.32 x=10-1 y=-1
13. [pic] 14. [pic] 15. [pic] 16. [pic]
3.79 -2 0.53 x=109
17. [pic] 18. [pic] 19. [pic] 20. [pic]
X=25/3 -4 undefined -6.14
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