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Change the following from exponential form to logarithmic form.
1) \[{3^4} = 81\]
2) \[{6^{ - 4}} = \dfrac{1}{{1296}}\]
3) \[{\left( {\dfrac{1}{{81}}} \right)^{\dfrac{3}{4}}} = \dfrac{1}{{27}}\]
4) \[{\left( {216} \right)^{\dfrac{1}{3}}} = 6\]
5) \[{\left( {13} \right)^{ - 1}} = \dfrac{1}{{13}}\]

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Last updated date: 20th Jun 2024
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Answer
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Hint:
Here we will use the basic concept of conversion of the exponential to the logarithmic form. In order to change from exponential form to logarithmic form, we need to identify the base of the exponential equation and move the base to the other side of the equal sign. Then add the word log to the expression formed.

Complete step by step solution:
We know that the exponential form is \[{a^x} = b\] and logarithmic form is \[x = {\log _a}b\].
Now we will apply this to all the options.
1) Given exponential form is \[{3^4} = 81\]
Now by using the basic concept of the conversion of the exponential into logarithmic function, we get
Logarithmic form is \[{\log _3}81 = 4\].

2) Given exponential form is \[{6^{ - 4}} = \dfrac{1}{{1296}}\]
Now by using the basic concept of the conversion of the exponential into logarithmic function, we get
Logarithmic form is \[{\log _6}\left( {\dfrac{1}{{1296}}} \right) = - 4\].
3) Given exponential form is \[{\left( {\dfrac{1}{{81}}} \right)^{\dfrac{3}{4}}} = \dfrac{1}{{27}}\]
Now by using the basic concept of the conversion of the exponential into logarithmic function, we get
Logarithmic form is \[{\log _{\dfrac{1}{{81}}}}\left( {\dfrac{1}{{27}}} \right) = \dfrac{3}{4}\].

4) Given exponential form is \[{\left( {216} \right)^{\dfrac{1}{3}}} = 6\]
Now by using the basic concept of the conversion of the exponential into logarithmic function, we get
Logarithmic form is \[{\log _{216}}6 = \dfrac{1}{3}\].

5) Given exponential form is \[{\left( {13} \right)^{ - 1}} = \dfrac{1}{{13}}\]
Now by using the basic concept of the conversion of the exponential into logarithmic function, we get
Logarithmic form is \[{\log _{13}}\left( {\dfrac{1}{{13}}} \right) = - 1\].

Note:
We should note that the value inside the log function should never be zero or negative; it should always be greater than zero. We need to remember that the value of the \[\log 10\] is equal to 1. The exponential function is a constant which is raised to some power. An exponential function is the inverse of the logarithmic function.
Some of the basic properties of logarithm is listed below:
 \[\begin{array}{l}\log a + \log b = \log ab\\\log {a^b} = b\log a\\\log a - \log b = \log \dfrac{a}{b}\\{\log _a}b = \dfrac{{\log b}}{{\log a}}\end{array}\]