Change of Base Formula with Problem Solved Example

If we are talking about the logarithms of a number then you are trying to find how many times a particular number should be multiple to get another number. For example, if you wanted to get 100 then multiple the ten twice or the logarithm of 100 with a base of 10 is 2. You can also take the help of a calculator to make things little easier for you. Log is the short form for writing logarithms in mathematics.

Further, let us discuss on base. What is base actually in mathematics? If we take the above example then base is 2 because 10 is multiplied twice. If you are using calculator then writing base is generally missing and logarithms are written in standard form only. One of the most important terms in logarithms is change of base formula that will give you the answer of a long with a different base by using log calculations with a standard base 10.

Change Of Base Formula

The change of base formula is the formula that will give you the answer of a log with a different base by using only log calculations with a base of 10. Here is standard change of formula in mathematics that can be given as below –

\[\LARGE \log_{b}x=\frac{\log _{d}x}{\log _{d}b}\]

Further, change of base formula helps in rewriting logarithms in terms of the base log. It can be used for evaluation of log with another base than 10. Practically, you can evaluate the non-standard base by transforming it to the fractions.

Change of Base Solve Example

Question 1: Solve \[\ 2 \log_{4} 29\]

Given \[\ 2 \log_{4} 29\]
Using logarithm change of base formula,

\[\ \log_{b}x=\frac{\log _{d}x}{\log _{d}b}\]
\[\ 2 \log_{4}29= 2 \times \frac{\log _{10}29}{\log _{10}4}\]
\[\ 2 \log_{4}29= 2 \times 2.43 = 4.86\]

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