What is the value of log2 128?
The Correct Answer and Explanation is:
ChatGPT said:
Answer:log2128=7\log_2 128 = 7log2128=7
Explanation:
To understand the value of log2128\log_2 128log2128, begin by recalling what a logarithm represents. The logarithmic expression logba=x\log_b a = xlogba=x means that bx=ab^x = abx=a. In this case, the base b=2b = 2b=2, and the number a=128a = 128a=128. The question asks: “To what power must 2 be raised to get 128?”
This can be solved by expressing 128 as a power of 2.
Start with repeated multiplication:21=222=423=824=1625=3226=6427=128\begin{align*} 2^1 &= 2 \\ 2^2 &= 4 \\ 2^3 &= 8 \\ 2^4 &= 16 \\ 2^5 &= 32 \\ 2^6 &= 64 \\ 2^7 &= 128 \end{align*}21222324252627=2=4=8=16=32=64=128
So, 27=1282^7 = 12827=128. That means 2 must be raised to the power of 7 to obtain 128. By the definition of logarithms:log2128=7\log_2 128 = 7log2128=7
This result makes sense because the logarithm is the inverse operation of exponentiation. When evaluating a logarithmic expression, the goal is to find the exponent that makes the base equal to the given number.
In general, if a number is a power of 2, then its base-2 logarithm is simply the exponent. Since 128=27128 = 2^7128=27, the base-2 logarithm of 128 is 7.
Logarithms have many applications in mathematics, science, and computer science. Base-2 logarithms, in particular, are useful in analyzing algorithms, especially in computer science, because binary systems operate using powers of 2. For example, the time complexity of certain efficient algorithms is expressed using base-2 logarithms.
Understanding logarithms helps with simplifying large computations, solving exponential equations, and analyzing growth or decay processes. The concept also plays a key role in fields such as signal processing, information theory, and cryptography.
