Understanding Logarithms

www.alexandertutoring.com Logarithms are mysterious to many students because they learn to get the answers by plugging numbers into their calculator. They often have very little intuitive understanding of what the logarithm really is. A logarithm is the inverse function of an exponential. Prior to learning logarithms there is no operation you can do to both sides of an exponential equation to isolate the variable. Students are used to dividing both sides by a number or subtracting a number from both sides of the equation. But when it comes to a variable in the exponent we have no method of isolating it. That's where logarithms come in to play. In order to understand what a logarithm really is I'd like us to go back in time to the 1700's when they were first introduced. Back then there were no calculators, so what did mathematicians do? Well they had tables of values in books. They had to look up the answers that way. Modern students are so used to using the calculator to evaluate logarithms that they have little intuitive understanding of what they really are. In this video I'm going to demonstrate how to construct the logarithm function by brute force. Meaning we will write down a number of exponential equations and solve each of them by guess and check. That is we will plug in numbers until we get the right answer. Once we put these guesses on a graph we will see they are connected, and the result is the logarithm function. The base of the logarithm corresponds to the base of the exponential you are trying to solve. So to solve an equation of the type 10 to the power of x you would use the logarithm base ten. To solve an equation of the type 5 to the power of x you would use the logarithm base 5 function, etc. I hope by seeing how the logarithm tables were constructed in the early days you will have a deeper appreciation for what the function really is, as opposed to a magic button in your calculator.