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The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: = + = Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition: [ 5 ] ln ( x ⋅ y ) = ln x + ln y ...
Undo is an interaction technique which is implemented in many computer programs. It erases the last change done to the document, reverting it to an older state. In some more advanced programs, such as graphic processing, undo will negate the last command done to the file being edited.
The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:
Control-Y is a common computer command. It is generated by holding Ctrl and pressing the Y key on most computer keyboards.. In most Windows applications this keyboard shortcut functions as Redo, reversing a previous Undo.
The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface.
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The logarithm is denoted "log b x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} .
The functions vary, however, according to the program you’re in. (For example, in an email program, F9 will check for new mail. In Microsoft Word, it will refresh your document.)