Question

What is the base conversion formula for logarithms?

Answer 1

Hint: Problems like these are quite easy to understand and are easy to solve once we understand the underlying concepts behind the question. For this particular problem, we need to have some fair amount of previous knowledge of fractions, logarithms and conversion from one logarithmic form to another. Here, we need to assume a constant on the basis of which we need to do the base conversion. Let us assume the constant here to be as ‘b’. We also need to assume a logarithm problem here to demonstrate the question. We assume, \[{{\log }_{c}}a\] . Here ‘c’ is the base of the logarithm and ‘a’ is known as the index.

Complete step-by-step answer:
Now we start off with the solution to the given problem by first of all writing the assumed problem here,
\[{{\log }_{c}}a\] . Now, we need to break this problem in such a way that, this problem is represented by a fraction of logarithms, such as \[\dfrac{{{k}_{1}}}{{{k}_{2}}}\] .
Now, by the formula of base conversion of logarithms, we can represent our logarithm problem as a fraction of two other logarithms. We write it as,
\[{{\log }_{c}}a=\dfrac{{{\log }_{b}}a}{{{\log }_{b}}c}\]
Here ‘b’ is our assumed constant which is also known as the base of the logarithm.

Note: For problems of these types we need to have some knowledge of logarithms and fractions. We need to be very careful about the conversion of the logarithm to quotient form as it may be a source of error to the problem. The assumptions in the problem need to be chosen according to the requirements to the question. When the base is less than that of the index, then it results in a negative value.