Unit 9.1 Introduction to Logarithmic Functions
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. They are how we “undo” exponential functions to solve for the exponents. Often it is shortened to “log” to symbolize the function
Rules of Logarithmic Functions
- logb (b) = 1
- logb (1) = 0
- logb (bx) = x
- [latex]b^{\log_b(x)}=x[/latex]
- logb (xn) = n logb (x)
Using the Rules of Logarithms to solve for x
Solve bx = n for x
[latex]b^x=n[/latex]
To “undo” or get rid of the b on both sides, take the log base “b” of both sides.
[latex]{log\ }_b\left(b^x\right)={\ log\ }_b\left(n\right)[/latex]
Per our 3rd rule, the exponent moves to the outside
[latex]x\ \cdot{log\ }_b\left(b\right)={\ log\ }_b\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{{log\ }_b\left(b\right)}={\ log\ }_b\left(n\right)[/latex]
Simplify
[latex]x={\ log\ }_b\left(n\right)[/latex]
So to convert between exponential form, and logarithmic form
exponential form logarithmic form
bx = n → logb (n) = x
Solving for x using logarithms
Solve for x : [latex]4^{x-2}=1[/latex]
Take the log base 4 of both sides
[latex]\log_4\left(4^{x-2}\right)=\log_4\left(1\right)[/latex]
Bring the exponent out front
[latex]\ x-2 \cdot\ log_4\left(4\right)=\log_4\left(1\right)[/latex]
Log4 (4) = 1 so it can be ignored
[latex]\ x-2 \cdot\cancel{log_4\left(4\right)}=\log_4\left(1\right)[/latex]
[latex]x-2=\log_4\left(1\right)[/latex]
Log (1) = 0
[latex]x-2=0[/latex]
x = 2
Common Log and Change of Base Formula
Most logs will have a defined base “b” but if it is not listed, then it is referred to as “common log” or assumed to have a base 10 in most applications. This is mainly to do with the fact that most modern numbering systems use base 10 (we have 10 fingers)
It will appear like this
log (x) = n
Today’s calculators can handle doing logarithmic functions of any base, but that wasn’t always the case. So someone would need to convert it to a base 10 before completing the function. they used the “change of base formula”
[latex]\log_b\left(n\right)=\frac{\log(n)}{\log(b)}[/latex]