Inverse Functions And Logarithms 2

 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

$${\log }_{b}x=y\iff b^y=x$$

For example:

$${\log }_2(8)=3$$ 

and that's mean 

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 

$$\ln (x)=y\iff e^y=x$$

$$\ln (e)=1$$

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

$$\ln (x)=\ln (e^y)$$

$$y\ln (e)=\ln (x)$$

$$y=\ln (x)$$

continued in part 3