Problem 67 - Algebra A Combined Function

Question

Evaluate each expression. $$ \log _{10} 100 $$

Step-by-Step Solution

Answer

\( \log_{10} 100 = 2 \)

Step 1: Understand the Expression

The expression \( \log_{10} 100 \) is a logarithmic expression, and it asks us to find the power to which the base 10 must be raised to get 100.

Step 2: Recall the Definition of Logarithm

A logarithm \( \log_{b} a \) answers the question: 'To what power must we raise the base \( b \) to get \( a \)?' Here, \( b = 10 \) and \( a = 100 \).

Step 3: Convert to Exponential Form

Convert the logarithmic equation \( \log_{10} 100 = x \) to its equivalent exponential form: \( 10^x = 100 \).

Step 4: Solve the Exponential Equation

We know that \( 10^2 = 100 \). Substitute 2 for \( x \) in the equation from Step 3. Therefore, the value of \( x \) is 2.

Key Concepts

Exponential Form

The exponential form is an essential concept that helps in understanding logarithms. Let's dive a little deeper into what this means. In mathematics, the exponential form expresses a number by showing how many times a base number is used as a factor. For example, if you see an expression like \(10^2\), it means you multiply 10 by itself to get 100. This expression is in exponential form because 10 is the base, and 2 is the exponent, indicating how many times 10 is used as a factor.

• Base: The number that is being multiplied by itself. In our example, it is 10.
• Exponent: The number that tells us how many times the base is multiplied. Here, it is 2.

Understanding exponential form is the key to working with logarithmic expressions because logarithms ask the reverse question: "To what power should the base be raised to get a certain number?" By converting between logarithmic and exponential forms, you can express logarithmic expressions in a way that might be easier to solve.

Base Ten Logarithms

Base ten logarithms are a specific type of logarithm where the base is 10. They are usually written as \( \log_{10} \) or simply \( \log \), where the base 10 is implicit. This type of logarithm is particularly useful because our number system is base 10, making these calculations common and relevant in practical applications. To understand a base ten logarithm, consider the question it answers: "To what power must 10 be raised to produce a given number?" For example, when you see \( \log_{10} 100 \), it asks, "What power should 10 be raised to get 100?"

• Simplified Notation: Often, you won't see the base explicitly written, and \( \log \) will be understood to mean \( \log_{10} \).
• Relation to Exponential Form: This question can also be seen in exponential form: what is \( x \) in \( 10^x = 100 \)?

Being efficient at working with base ten logarithms is a handy skill that helps in evaluating expressions quickly, understanding scientific notation, and even working in fields like data science and engineering.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves finding the power to which a given base must be raised to produce a specified number. When dealing with expressions like \( \log_{10} 100 \), you're essentially asking what exponent makes the base, in this case, 10, equal to 100 when raised to that power. To evaluate \( \log_{10} 100 \), you can follow a set series of steps:

• Convert to Exponential Equation: Express the logarithmic statement as an equation, \( 10^x = 100 \).
• Solve for the Exponent: Recognize that \( 10^2 = 100 \), so \( x = 2 \).

Once you have the exponent, you've evaluated the logarithmic expression. The answer to \( \log_{10} 100 \) is 2. This process becomes second nature with practice, allowing for quick evaluation of logarithmic expressions, whether they involve simple integers or more complex numbers. Understanding this evaluation process is crucial, especially because logarithms are ubiquitous in various scientific and technical calculations.