Problem 57 - Algebra 2

Question

Evaluate each expression. $$ \log _{7} 7^{2 x} $$

Step-by-Step Solution

Answer

The expression evaluates to \( 2x \).

Step 1: Recall the Logarithm Power Rule

The logarithm power rule states that \( \log_b(x^n) = n \cdot \log_b(x) \). This means you can bring down the exponent in front of the log.

Step 2: Apply the Power Rule to the Given Expression

Using the power rule, we can rewrite the given expression \( \log_7(7^{2x}) \) as \( 2x \cdot \log_7(7) \).

Step 3: Evaluate \( \log_7(7) \)

The logarithm \( \log_b(b) \) always evaluates to 1 because any number to the power of 1 is itself. Therefore, \( \log_7(7) = 1 \).

Step 4: Simplify the Expression

Substitute the value of \( \log_7(7) \) into the expression from Step 2. This gives \( 2x \cdot 1 = 2x \).

Step 5: Final Answer

The expression simplifies to \( 2x \). This is the simplest form of the given logarithmic expression.

Key Concepts

Logarithm Power Rule

The logarithm power rule is a handy tool that simplifies expressions where an exponent is involved. It allows you to move the exponent in a logarithmic expression to the front as a multiplier. For instance, if you have the expression \( \log_b(x^n) \), you can convert it to \( n \cdot \log_b(x) \). This simplification often makes it easier to evaluate or further manipulate logarithmic expressions.

• It reduces the complexity of the expression by eliminating the exponent.
• This rule relies on the understanding that exponents and logarithms have inverse operations.

Applying the logarithm power rule to the expression \( \log_7(7^{2x}) \), we move the exponent \( 2x \) in front as a multiplier, turning the expression into \( 2x \cdot \log_7(7) \). This transformation is the first step towards simplifying the logarithmic expression.

Logarithm Base Change

Changing the base of logarithms can help when the base of the log does not directly correspond with the numbers involved. Although simplifying the exercise does not require a base change, it's crucial to understand it for broader use. If you need to change the base of a logarithmic number from base \( a \) to base \( b \), you can use the formula:\[ \log_a(x) = \frac{\log_b(x)}{\log_b(a)} \] This formula is especially useful when using calculators since most provide logarithms in base 10 (common logarithm) or base e (natural logarithm).

• Base change is particularly helpful when your calculation tool does not support the current base.
• The rule ensures any logarithm can be transformed to a base that simplifies computation.

While only directly referenced here, base change knowledge often supplements fundamental logarithmic operations.

Logarithmic Expressions

A logarithmic expression is simply a mathematical statement involving logarithms. It often includes a base and an argument, such as \( \log_b(x) \), where \( b \) is the base and \( x \) is the argument. Understanding these components helps in manipulating expressions effectively.

• The base (e.g., \( b \)) tells us which number raised to a power gives the argument.
• The argument (e.g., \( x \)) is the number we're taking the log of, indicating the power that the base is raised to.

For the expression \( \log_7(7^{2x}) \), note how we simplified it using the power rule. First, the power rule allowed breaking down the expression to \( 2x \cdot \log_7(7) \).Then it became apparent how \( \log_7(7)\) equals 1, simplifying to \( 2x \). Understanding each element and how rules apply to them is crucial for solving and evaluating logarithmic expressions efficiently.