To solve the problem of rewriting the equation [tex]\( 81 = 3^4 \)[/tex] in logarithmic form, we need to understand the relationship between exponential and logarithmic forms. The exponential form of the equation defines how many times the base needs to be multiplied to achieve the result. Here, [tex]\( 3 \)[/tex] is the base, [tex]\( 4 \)[/tex] is the exponent, and [tex]\( 81 \)[/tex] is the result.
The logarithmic form of the equation expresses the exponent as the result of a logarithm operation. Specifically, the logarithm asks the question: "To what power must the base be raised to obtain the given result?"
Given the equation:
[tex]\[ 81 = 3^4 \][/tex]
To convert this to logarithmic form, we follow these steps:
1. Identify the base of the exponential equation. Here, the base is [tex]\( 3 \)[/tex].
2. Identify the result of the exponential equation. Here, the result is [tex]\( 81 \)[/tex].
3. Identify the exponent of the exponential equation. Here, the exponent is [tex]\( 4 \)[/tex].
Using these components, we rewrite the equation in logarithmic form. The logarithmic form asks, "To what power must the base [tex]\( 3 \)[/tex] be raised to get [tex]\( 81 \)[/tex]?"
Therefore, we write:
[tex]\[ \log_{3}(81) = 4 \][/tex]
So, the equation [tex]\( 81 = 3^4 \)[/tex] in logarithmic form is:
[tex]\[ \log_{3}(81) \][/tex]
And the value of the logarithm is:
[tex]\[ 4 \][/tex]
Thus, the complete logarithmic form with the result is:
[tex]\[ \log_{3}(81) = 4 \][/tex]
