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Exponentiation is a mathematical operation that involves taking a base number and raising it to a power. The expression for exponentiation is written as \(b^y\), where \(b\) is the base and \(y\) is the exponent. For example, \(2^3 = 8\) means that the base 2 is multiplied by itself three times to get 8.

Here are some key points about exponentiation:

• The base can be any real number.
• The exponent can be positive, negative, or zero.
• When the exponent is positive, the base is multiplied by itself. For instance, \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).
• When the exponent is zero, any nonzero base raised to the power of zero equals 1. For example, \(5^0 = 1\).
• When the exponent is negative, the result is the reciprocal of the base raised to the positive exponent. For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).

Exponentiation is fundamental in understanding logarithmic functions, as logarithms are inverses of exponentiation.

The base 2 logarithm, or binary logarithm, is the inverse operation of exponentiation with base 2. The mathematical notation is \(\log_2(x)\), which reads as 'log base 2 of x'. It means finding the exponent \(y\) such that \(2^y = x\).

For instance:

• For \(\log_2(8)\), since \(8 = 2^3\), it follows that \(\log_2(8) = 3\).
• For \(\log_2(32)\), since \(32 = 2^5\), it follows that \(\log_2(32) = 5\).

Here are some properties of the base 2 logarithm:

• \(\log_2(1) = 0\) because \(2^0 = 1\).
• \(\log_2(2) = 1\) because \(2^1 = 2\).
• \(\log_2(\frac{1}{8}) = -3\) because \(\frac{1}{8} = 2^{-3}\).

The base 2 logarithm is particularly useful in computing and information theory, where it helps to quantify concepts such as entropy and information content.

Function evaluation involves finding the output of a function given a specific input value. For the function \(f(x) = \log_2(x)\), we evaluate it by determining the exponent that the base 2 must have to equal the input \(x\).

Let’s go through some examples from the problem:

• For \(f(8)\), since \(8 = 2^3\), \(f(8) = 3\).
• For \(f(32)\), since \(32 = 2^5\), \(f(32) = 5\).
• For \(f(2)\), since \(2 = 2^1\), \(f(2) = 1\).
• For \(f\left(\frac{1}{8}\right)\), since \(\frac{1}{8} = 2^{-3}\), \(f\left(\frac{1}{8}\right) = -3\).

By evaluating the function \(f(x) = \log_2(x)\) in these examples, we see how logarithms transform multiplicative relationships into additive ones, simplifying the process of finding the exponent.

Inverse operations are pairs of mathematical operations that reverse the effect of each other. In the case of logarithms and exponentiation, these are inverse operations.

When you exponentiate a number, and then take the logarithm of the result, you essentially return to the original number. Mathematically, this is expressed as:

• \(\log_b(b^y) = y\)
• \(b^{\log_b(x)} = x\)

For example:

• \(\log_2(2^3) = 3\)
• \(2^{\log_2(8)} = 8\)

These properties highlight how logarithms and exponentiation undo each other. This relationship is vital for solving equations involving exponents and logarithms, as it provides a method to simplify and solve them. Understanding inverse operations clarifies the concept of solving for unknown exponents in logarithmic functions, making it easier to evaluate them effectively.