Change of Base Formula

Calculators only have buttons for log_{10}, written log, and log_e, written ln.

So how can we take log_2 using our calculators? If it’s hard we need to use our calculators!

We use the Change of Base Formula!!!

    \[log_{b}(a)=\frac{log_{c}(b)}{log_{c}(a)}\]

Example 1: log_6(7776)=\frac{log(7776)}{log(6)}=5

Proof of Change of Base

 

Given

    \[log_{a}(x)=y \Longleftrightarrow a^y=x\]

Want to find

    \[log_{b}(x)\]

Since

    \[x=a^y \Longrightarrow log_{b}(x)=log_{b}(a^y)\]

Power property

    \[log_{b}(x)=ylog_{b}(a)\]

Solving

    \[y=\frac{log_{b}(x)}{log_{b}(a)}\]