Problem 72 - Algebra A Combined Function

Question

Simplify. $$ 7^{\log _{7} 4} $$

Step-by-Step Solution

Answer

The expression simplifies to 4.

Step 1: Understand the Expression

The expression is given as \( 7^{\log_{7} 4} \). Here, we have a base of 7 and an exponent \( \log_{7} 4 \). Our task is to simplify this expression.

Step 2: Recognize the Logarithmic Identity

Recall the logarithmic identity \( a^{\log_{a} b} = b \). This identity tells us that if we have an exponent that is a logarithm with the same base as the main base, the expression simplifies directly to \( b \).

Step 3: Apply the Logarithmic Identity

Using the identity \( a^{\log_{a} b} = b \), apply it to our expression. Here, the base \( a \) is 7 and \( b \) is 4. Therefore, \( 7^{\log_{7} 4} = 4 \).

Key Concepts

Simplifying Expressions

Simpifying expressions in mathematics often involves using known rules or identities to rewrite an expression in a simpler form. In the given problem, simplifying the expression \( 7^{\log_{7} 4} \) means reducing it to a more straightforward form without changing its value.

The key to simplification is recognizing patterns or components, such as the presence of common bases and logarithmic properties. This often leads to reducing complex expressions into simpler numbers or terms. Here, we use a specific logarithmic identity to directly transform our complex expression into a basic number. This makes it easier to interpret and use.

Understanding simplification can save time and effort during problem-solving by reducing long or cumbersome calculations into manageable parts.

Exponent Rules

Exponent rules are fundamental in simplifying expressions and solving equations involving powers. The exponent in our exercise is \( \log_{7} 4 \), which immediately suggests using logarithmic rules alongside exponentiation concepts.

Basic rules for exponents include:

• Product of powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a power: \( (a^m)^n = a^{m \cdot n} \)
• Power of a product: \( (ab)^n = a^n b^n \)

These rules indicate how to handle operations involving exponents on the same base. For example, an exponent that is a logarithm with the same base suggests we can use a logarithmic identity to simplify the expression.

Recognizing when to apply these rules is key to manipulating and simplifying expressions efficiently.

Base and Exponent

When you see an expression like \( 7^{\log_{7} 4} \), it's crucial to identify the base and exponent. In this context, the base is 7, and the exponent is the logarithmic expression \( \log_{7} 4 \). Both components are essential in applying the appropriate simplification rule.

The base signifies the repeated factor in repeated multiplication, while the exponent denotes how many times the base multiplies itself.

When the exponent happens to be a logarithm that shares the base of the overarching power, it indicates a straightforward reduction is possible. In our example, we use the identity \( a^{\log_a b} = b \), which simplifies the '{base, exponent}' structure to just the value of the logarithmic identity outcome—in this case, 4.

Logarithms

Logarithms are the inverse operations of exponentiation and play a crucial role in handling exponential expressions. In the exercise, the expression \( \log_{7} 4 \) is a logarithm with base 7. This expression is the same as finding the power to which 7 must be raised to result in 4.

Logarithms follow specific rules that can simplify even the most complex expressions:

• \( \log_a (xy) = \log_a x + \log_a y \)
• \( \log_a \left( \frac{x}{y} \right) = \log_a x - \log_a y \)
• \( \log_a (x^y) = y \cdot \log_a x \)

These properties make logarithms powerful tools in both expansion and simplification tasks, permitting transformations into simpler equivalent forms.

Applying the logarithmic identity \( a^{\log_a b} = b \) makes simplifying expresssions like \( 7^{\log_7 4} \) straightforward by eliminating the logarithmic complexity and reducing it directly to the value 4. Understanding logarithms allows us to handle and transform expressions elegantly and efficiently.