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This Lesson (CHANGING THE BASE OF A LOGARITHM) was created by by Theo(672)   About Theo:
This lesson covers an overview of CHANGING THE BASE OF A LOGARITHM
REFERENCES Two major references for this lesson can be found at the following web addresses: http://www.sosmath.com/algebra/logs/log1/log1.html http://www.themathpage.com/aPreCalc/logarithms.htm#laws These covered the subject very well, but if you need more, just do a search on yahoo or google or any other search engine. VALUE OF CHANGING THE BASE OF LOGARITHMS Being able to change the base of a logarithm is essential to solving problems involving bases other than the bases used in your calculator. Most scientific calculators use the common logarithm to the base 10. That would normally be shown as the LOG key. Most scientific calculators use the natural logarithm to the base e. That would normally be shown as the LN key. Most scientific calculators do not calculate logarithms for bases other than 10 or e. There are some that do, but these are not anywhere near as common as the ones that don't. First some basics. DEFINITION OF A LOGARITHM This means that y equals the logarithm of x to the base b if and only if the base b raised to the power of the exponent y equals x. In other words, the logarithm of a number to a base, is the exponent that the base is raised to in order to get the number. EXAMPLE 1 This means that the log of 1000 to the base 10 equals the exponent 3 which is the power that the base 10 is raised to in order to get 1000. EXAMPLE 2 This means that the log of 148.4131591 to the base e equals the exponent 5 which is the power that the base e is raised to in order to get 148.4131591. EXAMPLE 3 This means that the log of 8 to the base 2 equals the exponent 3 which is the power that the base 2 is raised to in order to get 8. EXAMPLE 4 This means that the log of 337.5 to the base 15 equals the exponent 3 which is the power that the base 15 is raised to in order to get 337.5. FINDING THE LOG OF 336 TO THE BASE 10 You're in luck here because most scientific calculators can find the logarithm to the base 10. In my calculator, I enter the number 336 and then hit the LOG key. Answer is: 2.5216339277 Now that you know what the log is, you can translate the logarithmic form to the exponential form of the equation easily by using the basic definition of logarithms that states that You can easily confirm that this is true by using your calculator to find the result of FINDING THE LOG OF 336 TO THE BASE e. You're in luck here again because most scientific calculators can find the logarithm to the base e. In my calculator, I enter the number 336 and then hit the LN key. Answer is: 5.81711116 Now that you know what the log is, you can translate the logarithmic form to the exponential form of the equation easily by using the basic definition of logarithms that states that You can easily confirm that this is true by using your calculator to find the result of FINDING THE LOG OF 336 TO THE BASE 6 You probably just ran out of luck because most calculators will not be able to find the logarithm of a number other than the common logarithm base of 10 or the natural logarithm base of e. Fortunately, there is a conversion formula that will help you in this case. This formula will allow you to convert a logarithm of any base to the logarithm of any other base. More then likely you would want to convert your logarithm to the base 10 or the base e, because that would allow you to use your calculator to solve the equation. FORMULA FOR CONVERTING A LOGARITHM FROM ONE BASE TO ANOTHER BASE Let one base = b let another base = c This means that the log of x to the base b is equal to the log of x to the base c divided by the log of b to the base c. The numerator is The denominator is In other words, the conversion formula takes the log of the original number taken to the new base and divides it by the log of the original base taken to the new base. We'll start our problem again and apply this formula to see if it helps. FINDING THE OF LOG OF 336 TO THE BASE 6 Our calculator can't do this for us, but now we have the conversion formula that will allow us to convert from the base 6 to any other base of our choosing. I will convert to the base 10 first and then convert to the base e just to show you that they both work and that they will provide the same answer. CONVERTING THE LOG OF 336 FROM THE BASE 6 TO THE BASE 10 The conversion formula states: let b = 6 and let c = 10 and let x = 336 Conversion formula becomes: Now we can use the LOG function of our calculator to solve this equation. We can easily confirm that this answer is correct by using the basic definition of logarithms to get: You can easily confirm that this is true by using your calculator to find the result of CONVERTING 336 FROM THE BASE 6 TO THE BASE e The Conversion formula states: let b = 6 and let c = e and let x = 336 Conversion formula becomes: Now we can use the LN function of our calculator to solve: We can easily confirm that this answer is correct by using the basic definition of logarithms to get: You can easily confirm that this is true by using your calculator to find the result of It is not surprising that: and: Once we get to that point, they have to be the same because they both represent Regardless of what base you converted to (LOG or LN), you got the same answer. DERIVATION OF THE FORMULA FOR CHANGING THE BASES OF LOGARITHMS This derivation makes use of the fact that our conversion formula can be written as This means that any log to the base b is equal to any other log to the base c times a constant factor that we are calling k. As it turns out, the constant factor k equals The derivation of this uses the fact that This means that the log of b to the base b = 1. This is true if and only if Since anything to the first power is equal to itself, this is automatically true for all values of b. If we allow x to be equal to the value of b, then our conversion formula of Since If we solve for k, we get If we substitute the value of k in the formula of then we get which becomes SOLVING THE LOG OF 336 TO THE BASE 6 WITHOUT THE USE OF THE CONVERSION FORMULA If we did not know or remember the conversion formula, we could still solve the problem of finding the logarithm of 336 to the base 6 as follows: Solve for y in the equation: We know that this formula is true if and only if We also know that if a = b, then log(a) = log(b). We apply this fact to the exponential form of our equation of In the lesson on LOGARITHMS, you will learn that If we let b = 10 and a = 6, then this equation becomes Because of this, our original equation of If we divide both sides of this equation by Since our calculator can do common logs using the LOG function (logarithms to the base 10 are called common logs), we are able to solve this equation using our calculator. Note that use of this technique led us back to use of the conversion formula that we either forgot or didn't know. We could have done the same with natural logs using the LN function of the calculator (logarithms to the base e are called natural logs). If a = b, then log(a) = log(b) If a = b, then ln(a) = ln(b) What was required was to use the basic definition of logarithms to convert the logarithmic equation to it's exponential form and then take the log of it. For more information on logarithms, see the lesson on LOGARITHMS For more information on exponents, see the lesson on EXPONENTS and the lesson on EXPONENTIAL EQUATIONS FINDING THE VALUE OF THE BASE e You can look it up in your text or on the internet or you can use your calculator to show what the value is. Finding it on your calculator takes advantage of the fact that On my calculator I would: Hit the 1 key (enter the number 1 into my calculator) Hit the [2nd] key Hit the [LN] key The result of those operations displays 2.718281828. Your calculator might have a different procedure. Because the value of e is irrational (not the result of dividing one integer by another), then the value of e is more accurately represented as 2.718281828... where the 3 dots following the number represent the fact that they don't end there, but continue on. In this case they continue on and on and on ... For more on irrational numbers, see the lesson on RULES OF ALGEBRAIC AND ARITHMETIC OPERATIONS Questions or comments regarding this lesson can be directed to dtheophilis@yahoo.com This lesson has been accessed 1078 times. |
