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Unformatted text preview: Definition of Log Function The logarithmic function with base b , where b > 0 and b 1, is denoted by and is defined by if and only if IN OTHER WORDS  AND I CAN NOT STRESS THIS ENOUGH A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. This definition can work in both directions. In some cases you will have an equation written in log form and need to convert it to exponential form and vice versa. So, when you are converting from log form to exponential form, b is your base, Y IS YOUR EXPONENT, and x is what your exponential expression is set equal to. Note that your domain is all positive real numbers and range is all real numbers. Example 1 : Express the logarithmic equation exponentially. We want to use the definition that is above: if and only if . First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 5 in our problem  so our base is going to be 5. Next, let's figure out the exponent. This is very key, again remember that logs are another way to write exponents. This means the log is set equal to the exponent, so in this problem that means that the exponent has to be 3. That leaves 125 to be what the exponential expression is set equal to. Putting all of this into the log definition we get: *Rewriting in exponential form Hopefully, when you see it written in exponential form you can tell that it is a true statement. In other words, when we cube 5 we do get 125. If you had written as 5 raised to the 125th power, hopefully you would have realized that was not correct because it would not equal 3. Example 2 : Express the logarithmic equation exponentially. We want to use the definition that is above: if and only if . First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 7 in our problem  so our base is going to be 7. Next, let's figure out the exponent. This is very key, again remember that logs are another way to write exponents. This means the log is set equal to the exponent, so in this problem that means that the exponent has to be y. That leaves 49 to be what the exponential expression is set equal to. Putting all of this into the log definition we get: *Rewriting in exponential form Example 3 : Express the exponential equation in a logarithmic form. This time I have you going in the opposite direction we were going in examples 1 and 2. But as mentioned above, you can use the log definition in either direction. These examples are to get you use to that definition: if and only if . First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 6 in our problem  so our base is going to be 6....
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This note was uploaded on 02/18/2011 for the course MATH 101 taught by Professor Kindle during the Spring '11 term at South Texas College.
 Spring '11
 Kindle
 Algebra

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