Is it true that x=elogx for all real x ?
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Step-by-Step Solution
Step 1: Analyze the expression
We need to determine if the given equation x=elogx holds true for all real values of x. This involves understanding the properties and domains of the exponential and logarithmic functions.
Step 2: Recall the definition of logarithm
The logarithm logx (often written as lnx in calculus, implying base e) is defined as the inverse of the exponential function. Specifically, if y=logbx, then by=x. When the base is e, we have y=lnx⟺ey=x.
Step 3: Consider the domain of the logarithm
A crucial aspect of the logarithm function is its domain. The logarithm of a number is only defined for positive real numbers. This means that for logx to be a valid expression, x must be strictly greater than zero.
Step 4: Evaluate the expression based on the domain
Based on the definition of the logarithm, elogx is indeed equal to x, but only for values of x where logx is defined. This means the equality holds true only for x>0. It does not hold for all real x, because logx is undefined for x≤0.