What is the logarithm of \(0.01\) ? The logarithm of \(10^{-2}\) is the same. Why?

Logarithms are like a magical bridge between the operations of multiplication and exponentiation to addition and subtraction. To understand this sorcery better, let's define a logarithm. A logarithm answers the question: 'What power do we raise a certain base to, in order to obtain a specific number?' For example, when we have an equation like \( 10^x = 100 \), what we're looking for with the logarithm is the value of \(x\) that makes the equation true.

In simpler terms, if the number we're looking at is written as \( b^y = x \), then the logarithm of \(x\) with the base \(b\) is \(y\), which we write as \( \log_b(x) = y \). For the common logarithm, the base is assumed to be 10, and usually, we just write it as \( \log(x) \). So, in our original exercise asking for \( \log(0.01) \), we're essentially seeking the power to which 10 must be raised to get 0.01.

Logarithms have special properties that make them incredibly handy for simplifying calculations and solving equations. These properties stem directly from the way exponents behave. Now, let's look at some of these properties that were implicitly used in the exercise solution:

Equality of Logarithms

If \( 10^a = 10^b \), then it must be true that \( a = b \). This means that when the bases and the results are equal, the exponents (or logarithms) must be equal too. This property is the key to solving our exercise.

Product Rule

The logarithm of a product is the sum of the logarithms of the factors. In other words, \( \log_b(xy) = \log_b(x) + \log_b(y) \).

Quotient Rule

Conversely, the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator: \( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) \).

Power Rule

Lastly, the logarithm of a power allows us to bring the exponent down as a multiplier: \( \log_b(x^y) = y \cdot \log_b(x) \).

These properties become valuable tools for us to rewrite and solve logarithmic equations and can transform complex problems into manageable ones.

Moving to the realm where numbers are repeatedly multiplied by themselves, we explore exponents and bases. An exponent, such as \(x\) in \(10^x\), indicates how many times we use the base (in this case, 10) in a multiplication. Exponents and bases are the foundation of logarithms and they always work hand in hand.

For instance, a base of 10 is often used in scientific measurements because our number system is decimal. When seeing numbers like \(10^2\) which equals 100, or \(10^{-2}\) which is 0.01, it's all about how many times 10 is multiplied or divided. Positive exponents signify repeated multiplication, while negative exponents mean division, or reciprocal multiplication.

In our exercise, understanding that \(0.01\) is the same as \(10^{-2}\) reveals a crucial aspect of exponents: they make it possible to easily handle very large or very small numbers by showing us the power relationship between the base and the result. This capability is what makes logarithms such a key tool in various fields, including science, engineering, and finance, where they help in unravelling the mysteries of growth, decay, and many natural phenomena.