Identify the expression e \( \log _{2} 128 \) \( \log _{4} 128 \) \( \log _{16} 128 \) \( \log _{128} 16 \)

Certainly! Let's evaluate each of the given logarithmic expressions: 1. **\( \log_{2} 128 \)** - **Interpretation**: Find the exponent \( x \) such that \( 2^x = 128 \). - **Calculation**: \[ 2^x = 128 \\ 2^x = 2^7 \quad (\text{since } 128 = 2^7) \\ \Rightarrow x = 7 \] - **Result**: \( \log_{2} 128 = 7 \) 2. **\( \log_{4} 128 \)** - **Interpretation**: Find the exponent \( x \) such that \( 4^x = 128 \). - **Calculation**: \[ 4^x = 128 \\ (2^2)^x = 2^7 \quad (\text{since } 4 = 2^2 \text{ and } 128 = 2^7) \\ 2^{2x} = 2^7 \\ 2x = 7 \\ \Rightarrow x = \frac{7}{2} = 3.5 \] - **Result**: \( \log_{4} 128 = \frac{7}{2} \) 3. **\( \log_{16} 128 \)** - **Interpretation**: Find the exponent \( x \) such that \( 16^x = 128 \). - **Calculation**: \[ 16^x = 128 \\ (2^4)^x = 2^7 \quad (\text{since } 16 = 2^4) \\ 2^{4x} = 2^7 \\ 4x = 7 \\ \Rightarrow x = \frac{7}{4} = 1.75 \] - **Result**: \( \log_{16} 128 = \frac{7}{4} \) 4. **\( \log_{128} 16 \)** - **Interpretation**: Find the exponent \( x \) such that \( 128^x = 16 \). - **Calculation**: \[ 128^x = 16 \\ (2^7)^x = 2^4 \quad (\text{since } 128 = 2^7 \text{ and } 16 = 2^4) \\ 2^{7x} = 2^4 \\ 7x = 4 \\ \Rightarrow x = \frac{4}{7} \approx 0.5714 \] - **Result**: \( \log_{128} 16 = \frac{4}{7} \) **Summary of Results:** - \( \log_{2} 128 = 7 \) - \( \log_{4} 128 = \frac{7}{2} \) - \( \log_{16} 128 = \frac{7}{4} \) - \( \log_{128} 16 = \frac{4}{7} \)