Question

Find the value of $\log \dfrac{{100}}{{90}}$

Answer 1

Hint : Since the numeral inside the log function is a fraction we use the quotient rule of logarithm also we use the power rule of logarithm in this problem.

Complete step-by-step answer :
Here we have to find the value of $\log \dfrac{{100}}{{90}}$, but the base is not mentioned in the problem. We usually take the base as 10 when the base is not mentioned.
Logarithm to the base 10 is known as common logarithm or decimal logarithm.
In $\log \dfrac{{100}}{{90}}$ the numeral $\dfrac{{100}}{{90}}$ can be simplified and written as$\dfrac{{10}}{9}$.
Therefore we have $\log \dfrac{{100}}{{90}} = \log \dfrac{{10}}{9}$
When the fractions are present in the log function we have to use the quotient rule of logarithm which is given by, ${\log _b}\dfrac{M}{N} = {\log _b}M - {\log _b}N$
$$$$$ \Rightarrow {\log _{10}}\dfrac{{10}}{9} = {\log _{10}}10 - {\log _{10}}9$We know that always logarithm of a numeral same as the base value is 1, i.e.${\log _b}b = 1$Because Logarithmic function is nothing but the inverse of exponential function therefore we can convert every logarithmic function into exponential form. i.e.$ \eqalign{ & {\log _a}b = c \Leftrightarrow b = {a^c} \cr & \Rightarrow {\log _a}a = c \Leftrightarrow a = {a^c} \cr & \Rightarrow c = 1 \cr} $Therefore we can write${\log _{10}}10 = 1$.$\eqalign{
& \Rightarrow {\log _{10}}10 - {\log _{10}}9 = 1 - {\log _{10}}9 \cr
& = 1 - {\log _{10}}({3^2}) \cr} $The power rule of logarithm is given by${\log _b}({x^n}) = n{\log _b}x$Applying the power rule we have${\log _{10}}({3^2}) = 2 * {\log _{10}}3$Also we know that${\log _{10}}3 \approx 0.4771$$ \eqalign{ & \Rightarrow 1 - {\log _{10}}({3^2}) = 1 - 2 * {\log _{10}}3 = 1 - 2 * 0.4771 \cr & = 1 - 0.9542 = 0.0458 \cr} $Thus$\log \dfrac{{100}}{{90}} = 0.0458 \approx 0.046$**So, the correct answer is “$\approx 0.046$”.**

Note : In solving problems containing logarithmic functions we have to carefully observe the base of the logarithmic function. If the base is mentioned explicitly then we have to carry the same base value throughout solving the problem. If the base is not mentioned then take the base as 10, if natural logarithm $$\ln (x)$$is given then the base is$e$.