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Answer :
- Recognize the exponential equation $2^{x} = 128$.
- Apply the conversion from exponential to logarithmic form: $b^y = x \Leftrightarrow \log_b x = y$.
- Rewrite the equation as $\log_2 128 = x$.
- The logarithmic form of the equation is $\boxed{\log _2 128= x}$.
### Explanation
1. Understanding the Problem
We are given the exponential equation $2^{x} = 128$ and we want to rewrite it in logarithmic form. The relationship between exponential and logarithmic forms is that $b^y = x$ is equivalent to $\log_b x = y$. In our equation, the base is 2, the exponent is $x$, and the result is 128.
2. Converting to Logarithmic Form
Using the relationship $b^y = x \Leftrightarrow \log_b x = y$, we can rewrite $2^{x} = 128$ as $\log_2 128 = x$.
3. Final Answer
Therefore, the logarithmic form of the given exponential equation is $\log_2 128 = x$.
### Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, measuring the intensity of sound (decibels), and determining the pH of a solution in chemistry. They are also used in computer science to analyze the complexity of algorithms and in finance to calculate compound interest.
- Apply the conversion from exponential to logarithmic form: $b^y = x \Leftrightarrow \log_b x = y$.
- Rewrite the equation as $\log_2 128 = x$.
- The logarithmic form of the equation is $\boxed{\log _2 128= x}$.
### Explanation
1. Understanding the Problem
We are given the exponential equation $2^{x} = 128$ and we want to rewrite it in logarithmic form. The relationship between exponential and logarithmic forms is that $b^y = x$ is equivalent to $\log_b x = y$. In our equation, the base is 2, the exponent is $x$, and the result is 128.
2. Converting to Logarithmic Form
Using the relationship $b^y = x \Leftrightarrow \log_b x = y$, we can rewrite $2^{x} = 128$ as $\log_2 128 = x$.
3. Final Answer
Therefore, the logarithmic form of the given exponential equation is $\log_2 128 = x$.
### Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, measuring the intensity of sound (decibels), and determining the pH of a solution in chemistry. They are also used in computer science to analyze the complexity of algorithms and in finance to calculate compound interest.
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