These problems include the cause of the need for renormalization, the cause of the failure of unification, and the cause of all the point problems in both QED and GR. In my theoretical research over the last decade, I have come to the conclusion that many of the problems in QED and GR are caused by fundamental and long-standing disclarities in the calculus. These papers have only been up about a week, but I can already see the firestorm ahead. I have been prodded by recent events to add an introductory paragraph here, explaining why I would want to attack the calculus. In doing so, I will show the magnificent cheat in the current derivation of dln(x)/dx, embarrassing every living mathematician. (In fact, these properties are why we call these functions “natural” in the first place!)įrom these, we can use the identities given previously, especially the base-change formula, to find derivatives for most any logarithmic or exponential function.The Natural Log and of 1/x by Miles MathisĪbstract: I will show that the current derivative of the natural log and the current derivative of 1/x are both wrong. Look at some of the basic ways we can manipulate logarithmic functions: This means that there is a “duality” to the properties of logarithmic and exponential functions.
Take a moment to look over that and make sure you understand how the log and exponential functions are opposites of each other. In general, the logarithm to base b, written \(\log_b x\), is the inverse of the function \(f(x)=b^x\). Therefore, the natural logarithm of x is defined as the inverse of the natural exponential function: For example log base 10 of 100 is 2, because 10 to the second power is 100. When we take the logarithm of a number, the answer is the exponent required to raise the base of the logarithm (often 10 or e) to the original number. Remember that a logarithm is the inverse of an exponential. We'll see one reason why this constant is important later on.
The natural exponential function is defined as Review of Logarithms and Exponentialsįirst, let's clarify what we mean by the natural logarithm and natural exponential function. While there are whole families of logarithmic and exponential functions, there are two in particular that are very special: the natural logarithm and natural exponential function. In this lesson, we'll see how to differentiate logarithmic and exponential functions. Differentiating a Logarithm or Exponentialīy now, you've seen how to differentiate simple polynomial functions, and perhaps a few other special functions (like trigonometric functions).
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