Question

How do you write the equation \[{{6}^{-2}}=\dfrac{1}{36}\] into logarithmic form?

Answer 1

Hint: The given equation is to be converted into logarithmic form. We will be using the properties of logarithmic function. We know that, \[{{\log }_{b}}a=n\] which implies that \[{{b}^{n}}=a\]. We will then rearrange the given expression and substitute the values in the formula as stated before and we will find the equivalent expression using logarithmic function.

Complete step by step solution:
According to the given question, we are given a mathematical statement which we have to write in logarithmic form.
We will first have an outlook of logarithm function.
Logarithm function can be explained as the inverse function with respect to exponential function. Logarithm function is represented by \[\log \].
For example – logarithm of a number, say 10, with base as 10 is,
\[{{\log }_{10}}10=1\]
In the question, we saw logarithmic form.
So, logarithmic form refers to the representation using logarithm function with a particular base as well. That is, \[{{b}^{n}}=a\] can be represented as \[{{\log }_{b}}a=n\].
For example - \[{{3}^{2}}=9\] can be written as \[{{\log }_{3}}9=2\].
The expression in the question given to us is,
\[{{6}^{-2}}=\dfrac{1}{36}\]----(1)
Logarithmic form: \[{{\log }_{b}}a=n\]----(2)
From equation (1), we can write,
\[b=6\], \[a=\dfrac{1}{36}\], \[n=-2\]
Now, we will substitute the values in the logarithm form equation (2), we get,
\[{{\log }_{6}}\dfrac{1}{36}=-2\]
Therefore, \[{{6}^{-2}}=\dfrac{1}{36}\] in the logarithmic form is \[{{\log }_{6}}\dfrac{1}{36}=-2\].

Note: The logarithmic formula should be correctly interpreted with respect to the question asked. While substituting the values in the logarithm formula, care should be taken as in which is the base part and which is the exponential part. Also, the fractional remains as it is, since we are only asked to write it in the logarithmic form. If the fractional part 1/36 is supposedly written as 36, then the answer would not be -2 anymore rather it would be 2.