Is it true that $x=e^{\log x}$ for all real $x$ ?

Step 1: Analyze the expression

We need to determine if the given equation $x = e^{\log x}$ 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 $\log x$ (often written as $\ln x$ in calculus, implying base $e$) is defined as the inverse of the exponential function. Specifically, if $y = \log_b x$, then $b^y = x$. When the base is $e$, we have $y = \ln x \iff e^y = x$.

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 $\log x$ 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, $e^{\log x}$ is indeed equal to $x$, but only for values of $x$ where $\log x$ is defined. This means the equality holds true only for $x > 0$. It does not hold for all real $x$, because $\log x$ is undefined for $x \le 0$.

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