Lesson 3 - Robert Lindblom Math & Science Academy



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]

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]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic]=x 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic] 16. [pic]

17. [pic] 18. [pic] 19. [pic] 20. [pic]

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