Unit 9.2 Natural Logarithms
Natural Logarithms
Natural logs are the inverse of the function y = ex. From unit 8.3, “e” is the natural number, therefore, the inverse is the natural log. The rules for natural logs are the same as regular logs, except that the base is always “e”. Instead of writing “log” to denote the function, it is abbreviated as “ln”
Rules of Natural Logarithms
- ln (e) = 1
- ln (1) = 0
- ln (ex) = x
- [latex]e^{\ln_e(x)}=x[/latex]
- ln (xn) = n ln (x)
Using Natural Log to solve for x
Solve ex = n for x
[latex]e^x=n[/latex]
To “undo” or get rid of the e, take the natural log of both sides.
[latex]{ln\ }\left(e^x\right)={\ ln\ }\left(n\right)[/latex]
Per our 3rd rule, the exponent moves to the outside
[latex]x\ \cdot{ln\ }\left(e\right)={\ ln\ }\left(n\right)[/latex]
Our first rule says that the logarithm with a base b of b is equal to 1. So it can be ignored
[latex]x\ \cdot\cancel{{ln\ }\left(e\right)}={\ ln\ }\left(n\right)[/latex]
Simplify
[latex]x={\ ln\ }\left(n\right)[/latex]
So to convert between exponential form, and logarithmic form
exponential form natural logarithmic form
ex = n → ln (n) = x
Solving for x using natural logs
Solve for x : [latex]e^{x-2}=1[/latex]
Take the ln of both sides
[latex]\ln\left(e^{x-2}\right)=\ln\left(1\right)[/latex]
Bring the exponent out front
[latex]\ x-2 \cdot\ ln\left(e\right)=\ln\left(1\right)[/latex]
ln (e) = 1 so it can be ignored
[latex]\ x-2 \cdot\cancel{ln\left(e\right)}=\ln \left(1\right)[/latex]
[latex]x-2=\ln\left(1\right)[/latex]
ln (1) = 0
[latex]x-2=0[/latex]
x = 2