Logarithms
we studied the inverse of one-to-one functions and we know that exponential functions are one-to-one for that we called the inverse of the exponential function is Logarithmic function
\[{\log _b}x = y \Leftrightarrow {b^y} = x\]
For example:
\[{\log _2}(8) = 3\]
and that's mean 
Laws of Logarithms: If x and y are positve numbers, then
- \[{\log _b}(xy) = {\log _b}(x) + {\log _b}(y)\]
- \[{\log _b}(\frac{x}{y}) = {\log _b}(x) - {\log _b}(y)\]
- \[{\log _b}({x^r}) = r{\log _b}(x)\]
\[{\log _e}(x) = \ln (x)\]
so that's mean
\[\ln (x) = y \Leftrightarrow {e^y} = x\]
\[\ln (e) = 1\]
For example:
solve the equation 
solution:
take ln for both sides
\[\ln ({e^{5 - 3x}}) = \ln (10)\]
\[(5 - 3x)\ln (e) = \ln (10)\]
\[5 - 3x = \ln (10)\]
\[x = \frac{1}{3}(5 - \ln (10))\]
Another nice property is
Q: Find the inverse of 
solution:
\[y = {e^x}\]
change every x to y and every y to x
\[x = {e^y}\]
rewrite the eqution with respect to y
we should take ln for both sides
\[\ln (x) = \ln ({e^y})\]
\[y\ln (e) = \ln (x)\]
\[y = \ln (x)\]
Tags:
calculus