Answer :
- Use the logarithm property $\log_b a^c = c \log_b a$.
- Apply this property to the expression $\log 10^{-101}$ to get $-101 \log 10$.
- Since $\log 10 = 1$, the expression simplifies to $-101 \times 1 = -101$.
- Therefore, $\log 10^{-101} = \boxed{{-101}}$.
### Explanation
1. Understanding the problem
We are asked to find the value of $\log 10^{-101}$. The logarithm is base 10, so $\log x = \log_{10} x$.
2. Applying Logarithm Properties
We will use the logarithm property $\log_b a^c = c \log_b a$. Applying this property to the expression $\log 10^{-101}$, we get $-101 \log 10$.
3. Simplifying the expression
Since $\log 10 = \log_{10} 10 = 1$, the expression simplifies to $-101 \times 1 = -101$. Therefore, $\log 10^{-101} = -101$.
4. Final Answer
The value of $\log 10^{-101}$ is $-101$.
### Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of a solution (pH scale). Understanding logarithms helps us to work with very large or very small numbers more easily.
- Apply this property to the expression $\log 10^{-101}$ to get $-101 \log 10$.
- Since $\log 10 = 1$, the expression simplifies to $-101 \times 1 = -101$.
- Therefore, $\log 10^{-101} = \boxed{{-101}}$.
### Explanation
1. Understanding the problem
We are asked to find the value of $\log 10^{-101}$. The logarithm is base 10, so $\log x = \log_{10} x$.
2. Applying Logarithm Properties
We will use the logarithm property $\log_b a^c = c \log_b a$. Applying this property to the expression $\log 10^{-101}$, we get $-101 \log 10$.
3. Simplifying the expression
Since $\log 10 = \log_{10} 10 = 1$, the expression simplifies to $-101 \times 1 = -101$. Therefore, $\log 10^{-101} = -101$.
4. Final Answer
The value of $\log 10^{-101}$ is $-101$.
### Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of a solution (pH scale). Understanding logarithms helps us to work with very large or very small numbers more easily.
