91Ó°ÊÓ

A logarithm is a way to find the exponent needed to produce a certain number from a given base. For instance, \(\text{log}_{2}(8)\) asks the question: 'To what power must 2 be raised to result in 8?' The answer is 3 because \({2}^{3} = 8\). Logarithms are written as \(\text{log}_{b}(x)\), where \('b'\) is the base and \('x'\) is the result we are trying to achieve. Therefore, \(\text{log}_{2}(x)\) specifically uses the base 2.

An exponential function describes a situation where a constant base, such as 2, is raised to a variable exponent. For example, in \({2}^{y}\), the base is 2 and \('y'\) is the exponent. As \('y'\) changes, the result of \({2}^{y}\) changes. When \('y'\) is a positive number, \({2}^{y}\) grows very quickly. For negative values of \('y'\), the function \({2}^{y}\) gets closer and closer to zero but never actually reaches zero. This is why there is no real exponent \('y'\) that you could plug into \({2}^{y}\) to get 0. This explains why \(\text{log}_{2}(0)\) is undefined: the equation has no solution.

Logarithms have several important properties that make them useful in math and science:
- The \(\text{log}_{b}(1) = 0\) because any non-zero number raised to the power of 0 is 1.
- The \(\text{log}_{b}(b) = 1\) because any number raised to the power of 1 equals itself.
- For any positive numbers \('a'\) and \('b'\), \(\text{log}_{b}(a \times b) = \text{log}_{b}(a) + \text{log}_{b}(b)\).
- For a positive number \('a'\) and \('b'\), \(\text{log}_{b}(\frac{a}{b}) = \text{log}_{b}(a) - \text{log}_{b}(b)\).
- Logarithms have an inverse relationship with exponential functions, helping to solve equations involving exponentiation.
- Lastly, \(\text{log}_{b}(x)\) is only defined for positive values of \('x'\); this means if \('x'\) is zero or negative, the logarithm does not exist and is therefore undefined.