Question

How do you write the equation ${\log _7}\left( {\dfrac{1}{{2410}}} \right) = - 4$ into exponential form?

Answer 1

Hint:
According to given in the question we have to determine the exponential form of the equation ${\log _7}\left( {\dfrac{1}{{2410}}} \right) = - 4$ which is as mentioned in the question. So, first of all to determine the exponential form of the equation we have to use the formula to solve the logarithmic which is as mentioned below:
Formula used:
$
   \Rightarrow {\log _b}x = a \\
   \Rightarrow x = {b^a} \\
 $…………………..(A)
Now, we have to use the formula (A) as mentioned just above to determine the solution of the logarithmic function by placing all the values in the formula.
Now, to solve the obtained expression we have to use the formula which is as mentioned below:
Formula used:
$ \Rightarrow {x^{ - 1}} = \dfrac{1}{x}...............(B)$

Complete step by step solution:
Step 1: First of all to determine the exponential form of the equation we have to use the formula (A) to solve the logarithmic which is as mentioned in the solution hint.
Step 2: Now, we have to use the formula (A) as mentioned in the solution hint to determine the solution of the logarithmic function by placing all the values in the formula. Hence,
$ \Rightarrow {7^{ - 4}}$
Step 3: Now, to solve the obtained expression we have to use the formula which is as mentioned in the solution hint. Hence,
$ \Rightarrow {7^{ - 4}} = \dfrac{1}{{2401}}$

Hence, with the help of the formula (A) and (B) we have determined the exponential form which is$ \Rightarrow {7^{ - 4}} = \dfrac{1}{{2401}}$.

Note:
1) To determine the required exponential form of the given logarithmic expression it is necessary that we have to use the formula (A) which is as mentioned in the solution hint.
2) To solve the negative power of the integer obtained we have to use ${x^{ - 1}} = \dfrac{1}{x}$which is already mentioned in the solution hint.