How do you use the properties of logarithms to rewrite and simplify the logarithmic expression of \[\log \left( \dfrac{7}{100} \right)\]?

Hint: In this question we have been given with a logarithmic expression which we have to simplify using appropriate logarithmic properties. We will use the division to subtraction property of logarithm which is $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ on the term. We will then use the exponent property of logarithm which is written as $\log {{a}^{b}}=b\log a$ and simplify the expression to get the required solution.

Complete step by step solution:
We have the expression given to us as:
\[\Rightarrow \log \left( \dfrac{7}{100} \right)\]
We have the term in the form of $\log \left( \dfrac{a}{b} \right)$ therefore, on using the division to subtraction formula $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ on the expression, we get:
$\Rightarrow \log 7-\log 100$
 Now we know that $100$ can be written as ${{10}^{2}}$, therefore on substituting, we get:
$\Rightarrow \log 7-\log {{10}^{2}}$
Now the term $\log {{10}^{2}}$ is in the form of $\log {{a}^{b}}$ therefore, on using the exponent property $\log {{a}^{b}}=b\log a$, we get:
$\Rightarrow \log 7-2\log 10$
Now we know that $\log 10=1$ therefore, we get the expression as:
$\Rightarrow \log 7-2$, which is the form for the given expression.

Note: It is to be noted that the logarithm we are using has the base $10$, the base is the number to which the log value has to be raised to, to get the original term. This is also called the antilog of the number which is the logical reverse of taking a log.
The most commonly used bases in logarithm are $10$ and $e$ which has a value of approximate $2.713...$
Logarithm is used to simplify a mathematical expression, it converts multiplication to addition, division to subtraction and exponents to multiplication.