Problem 30 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \log _{6} 6^{9} $$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Understand the Logarithmic Identity

Recall the logarithmic identity \( \log_b(b^x) = x \). This identity tells us that when you have a logarithm of a number that is raised to a power, and the base of the logarithm is the same as the base of the exponent, the result is simply the exponent.

Step 2: Apply the Identity to the Expression

Notice that the expression \( \log_{6}(6^9) \) matches the identity from Step 1: the base of the logarithm (6) is the same as the base of the exponent (also 6). Therefore, according to the identity, the expression simplifies directly to the exponent.

Step 3: Simplified Result

Applying the identity from Step 1 to the given expression \( \log_{6}(6^9) \), we find that the expression simplifies to the exponent, which is 9.

Key Concepts

Logarithmic Identity

In the world of logarithms, simplifying expressions can often be achieved through understanding and applying key identities. One of the most important is the logarithmic identity: \( \log_b(b^x) = x \). This identity is useful whenever you have a logarithm where the base matches that of a number raised to an exponent.Let's break this down:

• The expression \( \log_b(b^x) \) means you're finding the power, \( x \), to which the base \( b \) is raised to get \( b^x \).
• If the base of the logarithm and the base of the power match, then the logarithm essentially cancels out, leaving you with the exponent \( x \).

This identity is a fantastic tool because it allows you to eliminate complex steps by directly simplifying the expression to its exponent, streamlining your calculations.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. It's expressed in the form \( b^x \), where \( b \) is the base and \( x \) is the exponent.Understanding basic properties:

• When you raise a base to an exponent, you're multiplying the base by itself as many times as the exponent indicates.
• For example, \( 6^9 \) means multiplying 6 by itself 9 times.

In the context of logarithms, knowing the base and exponent is crucial because it engages directly with the logarithmic identity. When both the logarithmic base and the exponential base are the same, as in \( \log_{6}(6^9) \), simplification becomes a lot more straightforward.

Simplifying Expressions

The process of simplifying mathematical expressions is about making the equation easier to work with by reducing it to its most basic form. When it comes to logarithmic expressions, simplification often involves applying identities like the one we've discussed.Why simplify?:

• It makes equations easier to understand and solve.
• It reduces computational complexity, saving time and effort.

In our specific example with \( \log_{6}(6^9) \), the simplification is all about recognizing the matching bases and applying the identity \( \log_b(b^x) = x \). This turns the original expression into the much simpler number 9, which is far more manageable for further calculations or interpretations.