Answer :

To change the base of a logarithm, you can use the change of base formula. The change of base formula allows us to convert a logarithm from one base to another by using common logarithms (base 10) or natural logarithms (base e).

Given the logarithm [tex]\log_5 212[/tex], we want to change the base to 12. According to the change of base formula, [tex]\log_{b} a = \frac{\log_{c} a}{\log_{c} b}[/tex], where [tex]c[/tex] is the new base you want to convert to.

In this case:

  1. [tex]a = 212[/tex], the number inside the original logarithm.
  2. [tex]b = 5[/tex], the original base of the logarithm.
  3. [tex]c = 12[/tex], the new base we want to use.

Applying the formula:

[tex]\log_{5} 212 = \frac{\log_{12} 212}{\log_{12} 5}[/tex]

This equation means that you can evaluate [tex]\log_5 212[/tex] by calculating [tex]\log_{12} 212[/tex] and [tex]\log_{12} 5[/tex] separately, and then dividing these two results.

To compute [tex]\log_{12} 212[/tex] and [tex]\log_{12} 5[/tex], you would typically use a calculator or software that supports logarithmic calculations in different bases. If that's not available, you might instead convert them using common logarithms (base 10), which most calculators support:

[tex]\log_5 212 = \frac{\log_{10} 212}{\log_{10} 5}[/tex]

Ultimately, understanding the change of base formula is crucial for solving logarithmic problems in different bases, and it can be used similarly to adapt to any desired base as part of your mathematical toolkit.