Problem 29 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(\log _{3} 3^{7}\)

Step-by-Step Solution

Answer

The value is 7.

Step 1: Identify the Logarithmic Property

Recognize that the expression represents a logarithmic function: \(\log _{a} a^{x} = x.\)This property states that if the base of the logarithm and the base of the exponent are the same, the result is simply the exponent.

Step 2: Apply the Logarithmic Property

Since the base of the logarithm and the base of the exponent are both 3, apply the property:\(\log _{3} 3^{7} = 7.\)

Key Concepts

Logarithmic Properties

Logarithmic properties are a set of rules and relationships that help us simplify and evaluate logarithmic expressions. These properties are crucial for manipulating expressions involving logarithms and making complex calculations much simpler. One fundamental property to understand is the product rule for logs:

• The product rule states that the logarithm of a product is the sum of the logarithms of the factors: \[\log_b(xy) = \log_b(x) + \log_b(y). \]


• There is also the quotient rule: The logarithm of a quotient is the difference of the logarithms: \[\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y). \]


• Finally, we have the power rule: The logarithm of a power is the exponent times the logarithm of the base: \[\log_b(x^a) = a \cdot \log_b(x). \]

These basic properties lay the foundations for working more comfortably with logs and can be applied to numerous problems involving different bases and arguments in the logarithmic world.

Evaluate Expressions

Evaluating expressions involving logarithms requires understanding the properties of logarithms and applying them effectively. In the context of logarithms, evaluation is about finding the numerical value of a given logarithmic expression. Let's consider evaluating \(\log_{3} 3^7\) as an example. The process can be broken down using the knowledge of properties:

• Identify Common Bases: Notice that both the base of the logarithm and the base of the exponent in the expression \(3^7\) are the same.


• Apply Logarithmic Property: By using the property \( \log_{a} a^x = x \), we see that \( \log_{3} 3^{7} \) can be directly evaluated to 7.

Evaluating expressions using logarithmic properties simplifies problems significantly. Understanding these mechanics allows one to swiftly solve problems without original cumbersome raw numeric expressions.

Exponents in Logarithms

The relationship between exponents and logarithms is foundational for understanding how logarithms work. Exponents and logarithms are inverse operations, much like multiplication and division. Here's how they are connected:

• For example, if you have \(b^y = x\), then \(\log_b(x) = y\). This expresses the idea that a base raised to a specific power gives a certain number.


• In our previous example of evaluating \(\log_{3} 3^7\), the exponent in the original exponentiated term 7 is the outcome of the logarithm application due to this inverse relationship.


• This property of exponents in logarithms helps in simplifying complex expressions, solving logarithmic equations, and it tells us precisely how powerful logarithms are when it comes to handling exponentiated numbers.

Recognizing this inverse relationship empowers you to traverse more challenging log problems with ease as the principles behind exponents directly inform and simplify the evaluation of logarithmic expressions.