Explanation:
The log formula rearranges an exponential term. I find it helpful to remember that a logarithm is an exponent.
If you start with the exponential term ...
[tex]b^x=a[/tex]
you will notice that the base of it is "b" and the exponent is "x". The log formula rearranges this to ...
[tex]\log_b{(a)}=x[/tex]
That is, "x" is the power to which the base must be raised in order to give you the value "a".
__
Examples
[tex]10^3=1000\\\\\log{(1000)}=3 \qquad\text{the base of 10 is assumed}[/tex]
When the base is e ≈ 2.7182818284590..., the logarithm is called the "natural log". Its function name is differentiated from the log base 10 by calling it "ln".
[tex]e^{-0.693147}\approx 0.5\\\\\ln{(0.5)}\approx -0.693147[/tex]
__
You know that multiplication is performed by adding exponents of powers of the same base:
[tex]10^5\times 10^3=10^{5+3}=10^8[/tex]
Logarithms work the same way that exponent arithmetic works:
[tex]\log{(10^5\times 10^3)}=\log{(10^5)}+\log{(10^3)}\\\\=5+3=8=\log{(10^8)}[/tex]
_____
e is an irrational number. One of the possible definitions of it is shown in the attachment.
