Identify the expression e \( \log _{2} 128 \) \( \log _{4} 128 \) \( \log _{16} 128 \) \( \log _{128} 16 \)
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To solve for each logarithmic expression, convert them into the same base when necessary. Starting with \( \log_{2} 128 \), we know that \( 128 = 2^7 \), so \( \log_{2} 128 = 7 \). Now, for \( \log_{4} 128 \), we rewrite it as \( \log_{4} (2^7) \). We can change the base to base 2 to get \( \frac{7}{2} \). Moving on to \( \log_{16} 128 \), since \( 16 = 2^4 \), we can express \( 128 \) as \( 2^7 \) and get \( \log_{16} (2^7) = \frac{7}{4} \). Finally, for \( \log_{128} 16 \), we have \( \log_{128} (2^4) \) which leads to \( 4/7 \) when converted. So the values are: 1. \( \log_{2} 128 = 7 \) 2. \( \log_{4} 128 = \frac{7}{2} \) 3. \( \log_{16} 128 = \frac{7}{4} \) 4. \( \log_{128} 16 = \frac{4}{7} \) These calculations show how logarithms can help simplify not just numbers, but understanding relationships between them! Understanding logarithms can feel daunting at first, but they're just another way to express exponentiation. For example, if you're trying to solve a real-world problem like determining how many times you need to double a specific amount of money to reach a target, logarithms are your best friend! They're everywhere in finance, physics, and even computer science.
