Question

You are supposed to write ${\log _7}49 = 2$ in exponential form.

Answer 1

We are given a logarithmic function which has a base of 7 and we have to write it in exponential form. For this we use the conversion of logarithms into exponents because both are inverse entities of each other.

Complete solution step by step:
Firstly we write the given logarithmic equation and name it as equation (1)
${\log _7}49 = 2$

Now we look what exponents actually mean

Exponent of a number means how many times the number is multiplied by itself i.e. ${p^q} = p \times p \times p \times p........(q\,{\text{times}}) = r$

It says $p$multiplied by itself $q$times equals to $r$

And logarithms are just opposite to it where the following function
${\log _a}c = b$

Means that - When $a$ is multiplied by itself $b$number of times $c$ is obtained.
This means equation (1) translates into 7 multiplied by itself twice (2 times) to obtain 49
$\Rightarrow 7 \times 7 = 49 \\\ \Rightarrow {7^2} = 49 \\\$
This is the exponential form of the given expression and this solution also helps us to derive an
important result which is
$[{\log _a}c = b] \Leftrightarrow [{a^b} = c]$

Additional information: Exponential function is Inverse function of a logarithmic function. This means one can be undone or removed by operating the other function on it and vice versa. This would give you a better understanding of it –
${\log _a}c = b \\\ {a^b} = {a^{{{\log }_a}\;c}} = c \\\$
Doing the opposite will give us –
${a^b} = c \\\ {\log _a}c = {\log _a}({a^b}) = b \\\$
This helps us to understand the reason why they are inverse functions with each other.

Note: Exponential functions are used to write large numbers into powers of 10 by using scientific notation and further, logarithmic functions are used to convert these exponents into smaller numbers and hence used a lot in real life as in measuring earthquakes, computing concentration chemicals etc.