- Recognize the definition: An exponential equation \(b^y = a\) is equivalent to \(\log_b a = y\).
- Identify the components in the given equation \(8^x = 64\), where base \(b=8\), exponent \(y=x\), and result \(a=64\).
- Substitute these components into the logarithmic formula to get \(\log_8 64 = x\).
- Match the result with the given options to find that the first choice is correct: \(\boxed{\log_8 64 = x}\).
Explanation
- Understanding the Relationship Between Exponential and Logarithmic Forms
To convert the exponential equation \(8^x = 64\) into its logarithmic form, we use the fundamental definition of logarithms. An exponential expression of the form \(b^y = a\) (where \(b > 0\) and \(b
eq 1\)) is equivalent to the logarithmic expression \(\log_b(a) = y\). In this relationship, \(b\) is the base, \(y\) is the exponent, and \(a\) is the result of the exponentiation.
- Identifying Components of the Equation
Given the equation \(8^x = 64\), we identify the corresponding components based on the standard form \(b^y = a\):
- The base \(b\) is \(8\).
- The exponent \(y\) is \(x\).
- The result \(a\) is \(64\).
- Rewriting in Logarithmic Form
By substituting our identified values (\(b=8\), \(y=x\), \(a=64\)) into the definition \(\log_b(a) = y\), we obtain the logarithmic equation:
\[\log_8(64) = x\]
Comparing this result to the provided options, we see that it perfectly matches the first choice.
- Conclusion
The correct logarithmic representation of the exponential equation \(8^x = 64\) is \(\log_8 64 = x\). Therefore, the correct option is the first one. \(\boxed{\log_8 64 = x}\)
Examples
Logarithms are used to solve for unknown exponents in real-world scenarios. For example, if you deposit money in a bank account that grows exponentially, you might use logarithms to calculate exactly how long it takes to reach a specific financial goal. By converting the growth equation into logarithmic form, you can determine the time variable, which is a practical application of understanding the inverse relationship between exponents and logarithms in finance, science, and data analysis.