Question

Convert the following into exponential form:
1) \[{\log _2}32 = 5\]
2) \[{\log _3}81 = 4\]

Answer 1

Hint:
Here, we are required to convert the given logarithmic forms into exponential forms. We will use their formula of conversion and substitute the given numbers in the formula to find the required exponential form of the given logarithmic forms. The logarithmic and exponential forms are inverses of each other. Exponential form means that a number is raised to a certain power.

Formula Used:
If there is a logarithmic function of the form \[{\log _a}y = x\], then in the exponential form it can be written as \[{a^x} = y\], where, \[a\] is the base, \[y\] is the number and \[x\] is the exponent

Complete Step by Step Solution:
To convert logarithmic form to an exponential form, first of all, we are required to find a base, which is the small number beside the ‘log’. The base always remains the base whereas the other two given numbers change their side. Hence, we will interchange the number and the exponent (which is present in the RHS) and remove the log sign to form the required exponential form.
\[{\log _2}32 = 5\]
We know that,
\[\begin{array}{l}{\log _a}y = x\\ \Rightarrow {a^x} = y\end{array}\]
Now, substituting \[a = 2\], \[y = 32\] and \[x = 5\] in the above conversion formula, we get
We get,
\[{\log _2}32 = 5\]
\[ \Rightarrow {2^5} = 32\]
Hence, \[{\log _2}32 = 5\] can be exponentially written as \[{2^5} = 32\].
\[{\log _3}81 = 4\]
We know that,
\[\begin{array}{l}{\log _a}y = x\\ \Rightarrow {a^x} = y\end{array}\]
Now, substituting \[a = 3\], \[y = 81\] and \[x = 4\] in the above conversion formula, we get
We get,
\[\begin{array}{l}{\log _3}81 = 4\\ \Rightarrow {3^4} = 81\end{array}\]

Hence, \[{\log _3}81 = 4\] can be exponentially written as \[{3^4} = 81\].

Note:
A logarithm of a number is the exponent that a base has to be raised to make that number. Using the conversion we can convert logarithms into exponential form and exponential to logarithmic form. Here, the base remains the same while the number and exponent changes around. Here we can make mistakes by writing base in place of exponent and exponent in place of base, and hence end up getting wrong answers.