Question
Write the exponential equation in logarithmic form. For example, the logarithmic form of \( 2^{3}=8 \) is \( \log _{2}(8)=3 \). \[ \begin{array}{l}32^{2 / 5}=4 \\ \text { Need Help? } \text { Read It } \text { Watch it }\end{array} \]
Ask by Wood Lynch. in the United States
Jan 23,2025
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Answer
The logarithmic form of \( 32^{\frac{2}{5}} = 4 \) is \( \log_{32}(4) = \frac{2}{5} \).
Solution
To convert the exponential equation \( 32^{\frac{2}{5}} = 4 \) into its logarithmic form, you can follow the example provided:
**Exponential Form:**
\[ 32^{\frac{2}{5}} = 4 \]
**Logarithmic Form:**
\[ \log_{32}(4) = \frac{2}{5} \]
**Explanation:**
- The base of the exponent (32) becomes the base of the logarithm.
- The result of the exponentiation (4) becomes the argument of the logarithm.
- The exponent itself (\(\frac{2}{5}\)) becomes the value of the logarithm.
So, the logarithmic form is:
\[ \log_{32}(4) = \frac{2}{5} \]
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Simplify this solution Beyond the Answer
To convert the exponential equation \( 32^{2/5} = 4 \) into logarithmic form, start by identifying the base, the result, and the exponent. The base is 32, the result is 4, and the exponent becomes the logarithm value. Therefore, the logarithmic form is \( \log_{32}(4) = \frac{2}{5} \). This transformation is crucial in understanding relationships between different forms of expressions. For instance, in mathematics, logarithmic and exponential forms offer various tools for solving equations and analyzing functions, making them both powerful and essential for higher-level learning!
