Question

How do you evaluate ${{\log }_{3}}7$ using the change of base formula?

Answer 1

Hint: The logarithm is used to convert a large or very small number into the understandable domain. For the theorem to work the usual conditions of logarithm will have to follow. We also need to remember that for logarithm function there has to be a domain constraint. The range in the usual case is the whole real line.

Complete step-by-step solution:
Let an arbitrary logarithmic function be $A={{\log }_{b}}a$. The conditions for the expression to be logical is $a,b>0;b\ne 1$.
We have $\log {{x}^{a}}=a\log x$. The power value of $a$ goes as a multiplication with $\log x$.
We also have the identity for logarithm where ${{\log }_{y}}x=\dfrac{{{\log }_{m}}x}{{{\log }_{m}}y}$.
The new base can be anything but the condition is that the base for both denominator and the numerator have to be the same.
Using the formula, we can break the logarithm ${{\log }_{3}}7$ taking the base as exponent $e$.
We have $\ln a={{\log }_{e}}a$.
Therefore, ${{\log }_{3}}7=\dfrac{{{\log }_{e}}7}{{{\log }_{e}}3}=\dfrac{\ln 7}{\ln 3}$.
Now we can put these values from the calculator to find the value for ${{\log }_{3}}7$.
We have $\ln 3=1.1,\ln 7=1.946$. This gives ${{\log }_{3}}7=\dfrac{\ln 7}{\ln 3}=\dfrac{1.946}{1.1}=1.098$.
The value of ${{\log }_{3}}7$ is $1.098$.

Note: There are some particular rules that we follow in case of finding the condensed form of logarithm. We first apply the power property first. Then we identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power. Then we apply the product property. Rewrite sums of logarithms as the logarithm of a product. We also have the quotient property rules.