TITLE: ${\log}_{a}{x}\neq {\int}^{x}_{1}{\frac{1}{t}}dt$
QUESTION [1 upvotes]: In most calculus textbooks, $\ln{x}$ is defined to be ${\int}^{x}_{1}{\frac{1}{t}}dt$. Some textbooks validate this definition by demonstrating that this function $\int^{x}_{1}{\frac{1}{t}}dt$ has all the properties of a logarithmic function (I've included pictures of this). I'm skeptical to this particular approach, since we could also define ${\log}_{a}{x}={\int}^{x}_{1}{\frac{1}{t}}dt$. We can still show that the laws of logarithms are properties of this integral, it's also obvious how the algebra will work out. And that means we've justified our claim?
Hell no! The derivative of ${\log}_{a}{x}$ is $\frac{1}{x}{\log}_{a}{e}$. Isn't this approach erroneous then? How then, could we show that this integral does not equal ${\log}_{a}{x}$? We could try showing that some of the properties of the logarithmic functions ($a\neq{e}$) do not hold for ${\int}^{x}_{1}{\frac{1}{t}}dt$. But how do we go about it?
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REPLY [1 votes]: Let us put it this way. Any continuous function $f:(0, \infty) \to\mathbb {R} $ with the property $$f(xy) =f(x) +f(y)$$ is said to be a logarithmic function.
If we start in this manner then there is no unique logarithmuc function. If $f$ satisfies the above property then so does any multiple of it. And there is a trivial function $f(x) =0$ which also works fine.
The key part is the following

Theorem: Let a function $f:(0, \infty) \to\mathbb {R} $ be continuous on $(0,\infty) $ and satisfy $f(xy) =f(x) +f(y) $. Then either $f(x) =0$ for all $x>0$ or there is a unique positive number $a\neq 1$ such that $f(a) =1$. The function $f$ is determined uniquely by the value of $a$. Let's use symbol $f_a$ for the function determined by $a$. Then $f_a$ is differentiable and further there is unique number $e>1$ with $f_e'(1)=1$ and more generally $$f_a(x) =\frac{f_e(x)} {f_e(a)}, \, f_a'(x) =\frac{1}{f_e(a)x}$$

The function $f_a(x) $ is called the logarithm of $x$ to the base $a$ and conventionally denoted by $\log_a x$. The function $\log_e x$ is simply denoted by $\log x$ or $\ln x$.
A proof of some parts of the theorem above is presented here. The introduction of the number $e$ and the fact that $f_a'(1)=1/f_e(a)$ can be proved with a little more effort and thus the formula in the theorem above can be established.
Also observe that $e$ can be introduced only when we consider the derivative of the logarithmiic function $f_a$. As long as we are working with algebraic properties we don't get to $e$. A similar treatment can be given for exponential functions using the property $f(x+y) =f(x) f(y) $.
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