Problem 54 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log 10^{2 x}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\log 10^{2x}\) is \(2x\).

Step 1: Identify the base and exponent

In our logarithmic expression, the base is 10 and the exponent is \(2x\). Hence the expression is similar to the form \(\log_b (b^m)\).

Step 2: Apply the logarithm property

We use the property \(\log_b (b^m) = m\). Here, \(b = 10\) and \(m = 2x\). Thus, \(\log 10^{2x} = 2x\).

Step 3: The final answer

The result of simplifying the expression \(\log 10^{2x}\) is \(2x\).

Key Concepts

Logarithm Properties

Logarithms have special properties that make them easier to work with, and these can help us simplify expressions neatly. One of the most useful properties is that of the common base:

• If you have a logarithm \(\log_b (b^m)\), it simplifies directly to the exponent \(m\).
• This property works because logarithms and exponents are inverse operations. The base \(b\) raised to the power of a logarithm with the same base \(\log_b\) will cancel out, leaving the exponent.

For example, with the expression \(\log 10^{2x}\), the logarithm and the base are both 10, leaving us only with the exponent \(2x\). This is why understanding these properties is so crucial for simplifying logarithmic expressions effectively.

Base and Exponent

Every logarithmic expression has a base and an exponent as its core components.

• **Base**: The base is the number that is raised to a power. It is crucial in defining what the logarithm is asking.
• **Exponent**: The exponent shows how many times the base multiplies by itself. It can be thought of as the number of iterations needed to reach a certain power number.

In the expression \(\log 10^{2x}\), the base is 10, and the exponent is \(2x\).
The exponent \(2x\) means that 10 is multiplied by itself \(2x\) times. Understanding this setup helps in applying logarithm properties correctly, ensuring we can simplify the expression to just the exponent.

Expression Simplification

Simplifying expressions, especially logarithm-based ones, involves unraveling complex structures to more straightforward forms.

• This is achieved by using properties of logarithms to cancel out terms, or to reduce them to more manageable numbers.
• By quickly identifying the base and exponent, we can apply logarithm properties directly, as seen in \(\log 10^{2x} = 2x\).

The goal in simplification is to reach the most reduced form, removing unnecessary components without altering the overall meaning.
Mastering simplification not only makes math problems easier but also builds foundational understanding, necessary for tackling more complex mathematical tasks in the future.