Inverse Functions And Logarithms 2

Published On :2021-10-17 06:37:00

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 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 {2^3} = 8



Laws of Logarithms: If  x and y are positve numbers, then
  1. [{log _b}(xy) = {log _b}(x) + {log _b}(y)]
  2. [{log _b}(frac{x}{y}) = {log _b}(x) - {log _b}(y)]
  3. [{log _b}({x^r}) = r{log _b}(x)]
Special Character: 
 [{log _e}(x) = ln (x)]
so that's mean 
[ln (x) = y Leftrightarrow {e^y} = x]
[ln (e) = 1]

For example:
solve the equation {e^{5 - 3x}} = 10
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 f(x) = {e^x}
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})]
[yln (e) = ln (x)]
[y = ln (x)]