There is a definition in my text that I don't really get the significance of. Could someone tell me what is the purpose of it?
Definition: The natural Logarithm function is the function for $x>0$ defined by $$\ln x=\int_{1}^{x} \frac{1}{t} dt.$$
Why is it $t$ (is it cause $x$ is already being used in the bounds)? What do those bounds have to do with $\ln$? From problems regarding this I see that I always have to make the functions bound start at 1 and end with $x$, why is that?
Thank You in advance!
$\endgroup$ 23 Answers
$\begingroup$What you are seeing is probably:
$\displaystyle \ln x := \int_1^x \frac1t \ \mathrm dt$
where $:=$ means "is defined as".
From the Fundamental theorem of calculus:
$\displaystyle \frac{\mathrm d}{\mathrm dx} \ln x = \frac 1x$
The lower bound is $1$ to make $\ln1=0$.
$\endgroup$ $\begingroup$By integrating, then evaluating at limits of integral and simplifying:
$$\int_1^x \frac{1}{t}dt=(\ln t)_1^x$$ $$=\ln x - \ln 1 = \ln x$$
$\endgroup$ $\begingroup$The following is the graph of $y=\frac{1}{x}$ showing the region bounded between the graph and the $x$-axis and bounded between vertical lines $x=1$ and $x=c$.
The area of the region is
$$\int_1^c\frac{1}{x}\,dx=\ln(c)\tag{1}$$
If one prefers to define the natural logarithm function in terms of $x$ rather than in terms of $c$, then one must use some other symbol than $x$ for the horizontal axis. Any other letter will be suitable, such as, for example, the letter $t$.
Then we write
$$\int_1^x\frac{1}{t}\,dt=\ln(x)\tag{2}$$
The issue is not to use one symbol $x$ to represent two different things. In equation $(1)$, $c$ is the constant and $x$ is the horizontal coordinate axis. In equation $(2)$, $x$ is the constant and $t$ is the horizontal coordinate axis.
