Problem 2 - Calculus

Question

Simplify the expressions completely. $$10^{\log (A B)}$$

Step-by-Step Solution

Answer

The simplified expression is \(AB\).

Step 1: Apply Logarithm Power Rule

Recalling from logarithm rules, the expression inside the logarithm, \(\log (A B)\), can be simplified using the multiplication rule of logarithms: \(\log (A) + \log (B)\). So, \(10^{\log (A B)} = 10^{\log A + \log B}\). This is the same due to the property of logs \(\log (AB) = \log A + \log B\).

Step 2: Use the Property of Exponential and Logarithms

Apply the property of exponential and logarithms where \(a^{\log_a (x)} = x\). Since the base of the logarithm \(10\) is the same as the base of the exponent \(10\), the expression can be simplified: \(10^{\log A + \log B} = A \cdot B\).

Key Concepts

Logarithm Rules

Logarithm rules are essential tools that help in simplifying and manipulating exponential expressions involving logarithms. Among these rules is the **product rule** that states:

• The logarithm of a product is equal to the sum of the logarithms of the individual factors.

For example, if you have an expression, \ \( \log (AB) \ \), it simplifies to \ \( \log A + \log B \ \).
This transformation is advantageous in breaking down complex expressions into more manageable parts, making it easier to simplify or solve equations.
If you face an expression like \ \( 10^{\log(A B)} \ \) and want to simplify it, start by recognizing that \ \( \log(A B) \ \) can be rewritten as \ \( \log A + \log B \ \), thanks to the power of logarithm rules.

Exponential and Logarithm Properties

Understanding the relationship between exponential and logarithm properties is crucial for simplifying expressions. One of the **key properties** to know is:

• When you have an exponent that matches the base of the logarithm, they can "cancel out" each other: \ \( a^{\log_a(x)} = x \ \).

This is because the exponentiation and logarithm are inverse operations.
For an expression like \ \( 10^{\log A + \log B} \ \), you can use this property if the base of the logarithm matches the base of the exponent. Here, base 10 is used for both the exponent and logarithm.
By applying this principle, you regain the product from the original logarithmic expression: \ \( 10^{\log A + \log B} = A \cdot B \ \). This simplification step leverages the inverse nature of exponential and logarithmic operations, simplifying the original complex expression into a basic product.

Multiplication Rule of Logarithms

The multiplication rule of logarithms is specifically used to handle log expressions that contain multiplications within them. It states:

• \ \( \log (AB) = \log A + \log B \ \)

This rule is quite straightforward yet powerful because it allows transformation of a product inside the log into a sum.
When you simplify an expression such as \ \( 10^{\log (AB)} \ \), the first step is applying the multiplication rule of logarithms. This leads to \ \( 10^{\log A + \log B} \ \).
This step exploits the fact that working with additions is often simpler than handling products, providing a neat path toward further simplification.
Overall, understanding this rule allows you to manipulate and simplify logarithm-based problems with greater ease, paving the way for solving a variety of real-world and theoretical mathematical problems.