Section. An Introduction to Logarithms So far we ve used the idea exponent Base Result from two points of view. When the base and exponent were given, for instance, we simplified to the result 8. When the exponent and result were given, for instance 8, we simplified to the base. A logarithm asks us to use the same idea from a third point of view. With a logarithm we re given a base and result and our job will be to find the exponent. In this section I ll introduce logarithms and show the relationship between logarithmic and exponential form... Some Vocabulary for Logarithms. A logarithm is the exponent a base must be raised to, to produce a result. The logarithm symbol, log, implies we are interested in finding the exponent. The original base, known as the base of the logarithm, is written a little lower and a little smaller log base. The original result is written to the right and is often called the argument log Argument. For example the expression log 8 is asking you to start with a base base of and find an exponent that returns a result of 8 so log 8. Definition: Logarithm English: The exponent a base must be raised to, to get a given result. Example: log 8 Note: The base should be greater than 0 and not equal to... Logarithmic and Exponential Form After finding the logarithm you can rewrite your work in exponential form to verify your answer. Definition: The Logarithm Form, Exponential Form Equivalence English: A logarithmic relationship can be written in exponential form. Example: log For instance earlier we found that log 8. Rewriting this equation in exponential form as 8 can help verify the answer was correct. Practice.. Logarithmic and Exponential Form a) log log First state the question the expression is asking. Then simplify the expression. Finally, rewrite your work in exponential form and simplify to check. log What exponent of two gives a result of sixteen? log Simplified the expression. Rewrote in exponential form to verify the work. log Five to what power gives a result of five? log Simplified the expression Rewrote in exponential form to verify the work. Copyright 0 Scott Storla. An Introduction to Logarithms
Homework. First state the question the expression is asking. Then simplify the expression. Finally, rewrite your work in exponential form and simplify to check. ) log ) log 000 ) 0 Log 8 ) log ) log ) log.. The Common and Natural Logarithm A scientific calculator has two different keys for logarithms. The common log key log implies the base is 0 so log is log. Base 0 is useful because our number system is a base 0 system. The natural 0 log key ln is used when the base is e so ln is log e. Recall that e is an irrational constant with a value approximately equal to.8. If you re lucky enough to study a first course in calculus you may learn why the name natural was chosen when discussing logarithms with a base of e. There s no agreement by calculator manufacturers on the order of the buttons you push to find log or ln. On some calculators you put in your argument and then push your log or ln key. On others you push the log or ln key first and then put in the argument. Try finding log00. Your calculator should say. Now try to find ln00. Your calculator should say.0 if you round to the ten-thousandths place. To check your work using exponential form you ll often use the power key or the if log00 use the power key to see if0 00. To check if ln00.0 use the x e key. To check x e key to see if.0 e 00. The wavy equals sign is necessary because.0 is only approximately equal to the real exponent (remember we rounded to the thousandths place). Since e.0 00.00, which is very close to 00, you should be comfortable with the answer. Practice.. The Common and Natural Logarithm a) log ln Simplify using your calculator. Rewrite your work in exponential form and simplify to check your work. 0.8 0.0000 0.8 The approximate answer from the calculator Rewrote in exponential form and used the calculator to check the work. 0.9 Rounded the answer. e 0.9 0. Checking the exponential form confirms the work. Homework. Simplify using your calculator. Check your answer. ) ln 8) log 9) log 0) ln ) log 9 ) ln. In some books log stands for log e. It won t in this course, but it s something you should be aware of. Copyright 0 Scott Storla. An Introduction to Logarithms
.. The Change of Base Formula Often calculators only have keys for log or ln. If your original base isn t 0 or e you can still use your calculator as long as you know the change of base formula. Procedure - The Change of Base Formula Using ln. To change a logarithmic expression to an equivalent expression with base e, build a fraction. The numerator is the natural log of the original argument. The denominator is the natural log of the original base. Example: log ln9 9 ln Practice.. The Change of Base Formula Simplify using the change of base formula. When appropriate round to the tenthousandths place. Use exponential form to check your answer. a) log 9 ln9 ln Used the change of base formula with base e. Simplified. Using log9 log would have given the same answer. log 00 log00 log Used the change of base formula with base 0..9 Simplified using a calculator. Rounded to the ten-thousandths place..9 00.00 Checked the answer. Homework. Simplify using the change of base formula. When appropriate round to the tenthousandths place. Use exponential form to check your answer. ) log 0.0 ) log 8) ) 9 log ) log,000 log. ) 8 log 8 0... Logarithms and the Order of Operations Simplifying logarithms is included in line of the order of operations. Also, the argument of a logarithm is a type of implicit grouping. Practice.. Logarithms and the Order of Operations a) ln8,000 /,000 Count the number of operations, name the operations using the correct order and then simplify the expression. If necessary round to the thousandths place. ln(.) Divided. There are two operations. The division would be first and then taking the natural logarithm. 0. Took the natural logarithm. Copyright 0 Scott Storla. An Introduction to Logarithms
log. There are three operations. Taking log base 0 of fifteen would be first, then multiplication by and finally division by Took log of and rounded. 0. Multiplied by and divided the result by. Homework. Count the number of operations, name the operations using the correct order and then simplify the expression. Only use a calculator when necessary. 9) ln 0) log9 ) ln 80 0 ) 00,000 ) ln 0.0 00,000 log0 log0 ) ln 8,000 0.0,000 ) ln... ) ln 0. 8) ) ln ln ln 0,000 ln,00 9) 0 ln 0 0) ln ln ln Copyright 0 Scott Storla. An Introduction to Logarithms
Homework. ) Six to what power gives a result of thirty-six?,, ) What exponent of ten gives a result of one thousand?,, ) Three to what power results in eighty-one?,, ) What power of four gives a result of four?,, 8 ) What exponent of six results in two hundred sixteen?,, ) Two to what power gives a result of thirty-two?,, 0,000 ) 0. 8).09 9).8 0) 0 ) 0. ) 0.8 ).9 ) ) 0.08 ) ) 8).8 9) There are two operations. First multiply and then take the logarithm. The answer is about.0. 0) There are two operations. First add then take the logarithm. The answer is about.9. ) There are five operations. First square the base of 0, next find the logarithms left to right, then multiply by and add last. The answer is. ) There are four operations. The division is first, the logarithm is next, followed by the exponent of negative one and then the multiplication. The answer is 9.. ) There are three operations. The division inside the parentheses is first, the logarithm is next and the division is last. The answer is about 0.09. ) There are four operations. The division is first, the logarithm is next, followed by the exponent of negative one and then the multiplication. The answer is about.. ) There are three operations. First simplify the argument by multiplying and then subtracting. Then take the logarithm. The answer is about 0.98. ) There are four operations. The division is first, the logarithm is next, followed by the exponent of negative one and then the multiplication. The answer is about.0. ) There are six operations. The division in the argument is first and then the logarithms and subtraction inside the parentheses. The logarithm outside the parentheses follows and the subtraction is last. The answer is 0. 8) There are three operations. The division inside the parentheses is first, the logarithm is next and the division is last. The answer is about 0.09. 9) There are five operations. First subtract and divide to simplify the argument. Then take the logarithm and multiply the quotient of one-tenth to the result. The answer is about 0.0. 0) There are five operations. First take the natural logarithms left to right and then add and subtract left to right. The answer is 0. Copyright 0 Scott Storla. An Introduction to Logarithms