Problem 81 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \log 100 $$.

Step-by-Step Solution

Answer

The answer is 2.

Step 1 - Basic Concept

Understand that \(\log a\) is asking the question: 10 should be raised to what power to obtain 'a'? In this particular exercise, \(\log 100\) is asking 10 to what power equals 100.

Step 2 - Evaluate

Here, since \(10^2 = 100\), the answer to \(\log 100\) is 2.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are the inverse operations of exponentiation. In essence, while an exponent says 'how many times do we multiply a number by itself to get another number,' a logarithm asks 'how many times do we need to multiply a certain base to get another number?' For instance, the expression \( \log 100 \) specifically means 'to what power must we raise 10 to get 100?' Understanding this inverse relationship is key to grasping logarithms.

Logarithms have a base, which in many common cases like this exercise is assumed to be 10 if no base is displayed. This is called the common logarithm. However, logarithms can be based on any positive number, not just 10, and these are typically written as \( \log_b a \) where \( b \) is the base and \( a \) is the number we're considering. Recognizing the base is the first step in comprehending any logarithmic expression.

Evaluating Logarithms

To evaluate logarithms, we apply the definition which relates directly to the concept of powers. For example, \( \log 100 \) is asking for the power that 10 must be raised to in order to result in 100. You can find this power by looking for a number, which when used as an exponent for the base, gives the original number. In the given exercise, \( 10^2 = 100 \) so, \( \log 100 = 2 \).

It's often helpful to remember a few common logarithms, such as \( \log 1 = 0 \) because \( 10^0 = 1 \) and \( \log 10 = 1 \) because \( 10^1 = 10 \). This way, evaluating logarithms can become more intuitive over time, and you won't always have to calculate each one from scratch.

Properties of Logarithms

Understanding the properties of logarithms can greatly simplify the process of calculating them. One key property is the 'product rule', which states that \( \log_b(xy) = \log_b(x) + \log_b(y) \), meaning that the log of a product is equal to the sum of the logs of its factors.

The 'quotient rule', on the other hand, tells us that \( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) \), indicating the log of a division equals the difference between the logs of the numerator and denominator. Another vital property is the 'power rule', which allows us to bring exponents out in front of the log: \( \log_b(x^y) = y \cdot \log_b(x) \).

Example Utilizing Logarithm Properties

These properties can transform complex logarithmic expressions into simpler ones. For instance, for evaluating \( \log(\frac{1000}{10}) \) without a calculator, you could use the quotient rule to see that this is the same as \( \log(1000) - \log(10) \) which simplifies to \( 3 - 1 = 2 \), because \( 10^3 = 1000 \) and \( 10^1 = 10 \). By mastering these properties, working with logarithms becomes much more manageable.