M1410303 - 3.18 SECTION 3.3: (MORE) PROPERTIES OF LOGS PART...

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3.18 SECTION 3.3: (MORE) PROPERTIES OF LOGS PART A: READING LOG EXPRESSIONS We will use grouping symbols as a means of clarifying the order of operations in expressions. Often, grouping symbols are omitted when they could have helped. How do we read log expressions in those cases? We use ln for convenience, but any log function with any nice base is dealt with similarly. Example: Read ln3 x + 7 as: ln 3 x ( ) + 7 Example: Read ln3 x 4 as: ln 3 x 4 ( ) Example: Read ln x 4 as: ln x 4 ( ) , not as ln x ( ) 4 Generally speaking, absent any grouping symbols, if we see “ln” or “log” followed by a product, quotient, power, or mixture of the above, the “ln” applies to the whole expression that follows. However, + and signs that introduce new terms tend to terminate the ln expression.
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PART B: LOG PROPERTIES BASED ON LAWS OF EXPONENTS Remember, logs are exponents. The laws for exponents imply laws for logs. We will state these properties using ln, though they apply to any log function with a nice base. If we use the rules from left-to-right, we are “expanding” the expression. If we use the rules from right-to-left, we are “condensing” the expression. Assume A > 0 and B > 0 . A and B may represent constants or variable expressions. Product Rule ln AB ( ) = ln A + ln B Think: The log of a product equals the sum of the logs. (We go one step down in the order of operations if we read left-to-right.) Related Exponent Law: When multiplying powers of e , the exponent on the product equals the sum of the exponents (of the factors): e A e B = e A + B Quotient Rule ln A B = ln A ln B Think: The log of a quotient equals the difference of the logs. Related Exponent Law: When dividing powers of
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M1410303 - 3.18 SECTION 3.3: (MORE) PROPERTIES OF LOGS PART...

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