Question

How do you evaluate ${\log _{20}}0.125$ using the change of base formula?

Answer 1

Hint: Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than $10$ or $e$ we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs. To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.

Formula used: According to change of base formula
${\log _a}b = \dfrac{{{{\log }_x}b}}{{{{\log }_x}a}} = \dfrac{{\log b}}{{\log a}}$

Complete step-by-step solution:
But when we use base $10$ we do not use subscript but for $20$ we have to write it in subscript
Hence
 $ \Rightarrow {\log _{20}}0.125 = \dfrac{{\log 0.125}}{{\log 20}}$
We can change the fraction to decimal point in the numerator and separate the values in denominator, therefore we get
$ \Rightarrow {\log _{20}}0.125 = \dfrac{{\log \left( {\dfrac{1}{8}} \right)}}{{\log \left( {2 \times 10} \right)}}$
Now in log we can write multiplication in the form of addition and division in the form of subtraction
Therefore, we can write is as
$ \Rightarrow \dfrac{{\log 1 - \log 8}}{{\log 2 + \log 10}}$
The value for $\log 1 = 0$ and $\log 10 = 1$, substituting that on above equation we get
$ \Rightarrow \dfrac{{ - \log {2^3}}}{{\log 2 + 1}}$
By taking $' - '$ as common, we can write it as
\[ \Rightarrow - \dfrac{{3\log 2}}{{1 + \log 2}}\]
Now substituting the value of $\log 2$, we get
$ \Rightarrow - \dfrac{{3\left( {0.301} \right)}}{{1 + 0.301}}$
Now solving the above terms by multiplication and addition, we get
$ \Rightarrow - \dfrac{{0.9042}}{{1.301}}$
On dividing them we get
$ \Rightarrow - 0.6950$

Hence using the change of base formula the value of ${\log _{20}}0.125$ is $ - 0.6950$

Note: The change-of-base formula can be used to evaluate a logarithm with any base.
For any positive real numbers $M,b$ and $n$ where $n \ne 1$ and $b \ne 1$
$ \Rightarrow {\log _b}M = \dfrac{{{{\log }_n}M}}{{{{\log }_n}b}}$
It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.
$ \Rightarrow {\log _b}M = \dfrac{{\ln M}}{{\ln b}}$ and
$ \Rightarrow {\log _b}M = \dfrac{{\log M}}{{\log b}}$
We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms.
We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms.
We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base.
We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input.
The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm.
We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula.
The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate.