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?