ࡱ> ')&y'` bjbj 4: %    "8>L<"#"s s s 1#3#3#3#3#3#3#$h$h&ZW#s s s s W#  l#"""s j ^1#"s 1#""J" pK3 "1##0#"*'w!*'"*'"8s s "s s s s s W#W#"js s s #s s s s """""""""       MA 15200 Lesson 29 Section 4.3 This lesson is on the properties of logarithms. Properties of logarithms model the properties of exponents. I Product Rule Product Rule of Exponents:  EMBED Equation.DSMT4  Notice: When the bases were the same, the exponents were added when multiplication was performed. Likewise logarithms are added when multiplication is performed in the argument.  Product Rule of Logarithms:  EMBED Equation.DSMT4  In words, the logarithm of a product is the sum of the logarithms. When a single logarithm is written using this product rule, we say we are expanding the logarithmic expression. Ex 1: Assume all variables represent positive values. Use the product rule to expand each expression and simplify where possible.  EMBED Equation.DSMT4  II Quotient Rule Quotient Rule for Exponents:  EMBED Equation.DSMT4  Notice: When the bases were the same, the exponents were subtracted when division was performed. Likewise, logarithms are subtracted when division is performed in the argument.  Quotient Rule for Logarithms:  EMBED Equation.DSMT4  In words, the logarithm of a quotient is the difference of the logarithms. We can also expand a logarithm by using the quotient rule. Ex 2: Assume all variables represent positive values.  Use the quotient rule to expand each logarithm and simplify where possible.  EMBED Equation.DSMT4  III Power Rule Power Rule for Exponents:  EMBED Equation.DSMT4  Note: When a power is raised to another power, the exponents are multiplied. Likewise, when a logarithm has an exponent in the argument, the exponent is multiplied by the logarithm. Power Rule for Logarithms:  EMBED Equation.DSMT4  In words, the logarithm of a power is the product of the exponent and the logarithm. We can also expand a logarithm by using the product rule. Ex 3: Assume all variable represent positive values. Use the power rule to expand each logarithm and simplify where possible.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  IV Here is a summary of all the properties of logarithms.  Ex 4: Use the properties to expand each logarithmic expression. Assume all variables represent positive values.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  In opposite of expanding a logarithmic expression is condensing a logarithmic expression. This is writing a logarithmic expression as a single logarithm. Ex 5: Condense each expression. In other words, write as a single logarithm. Assume all variables represent positive values.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.DSMT4  Ex 6:  EMBED Equation.DSMT4 , use the properties of logs to find the following values.  EMBED Equation.DSMT4  Ex 7: If  EMBED Equation.3 . Use these values and the properties of logs to find the following values.  EMBED Equation.DSMT4    EMBED Equation.3  Ex 8: Let  EMBED Equation.DSMT4 . Write each expression in terms of A and/or B.  EMBED Equation.DSMT4  V Change of Base Formula Your scientific calculator will approximate or find common logarithms (base 10) or natural logarithms (base e). How can logarithms with other bases be approximated?   EMBED Equation.DSMT4  The formula above is known as the change of base formula. Ex 9: Approximate each logarithm to 4 decimal places.  EMBED Equation.DSMT4      PAGE  PAGE 5 Informal Proof:  EMBED Equation.DSMT4  CAUTION:  EMBED Equation.3  Note: Our text and online homework does not usually use parenthesis around the argument. However, it would be better to write as in the following.  EMBED Equation.DSMT4  Assume all variables represent positive values and that all bases are positive number (not 1).  EMBED Equation.DSMT4  There is more than 1 way to determine these values.  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