Problem 30 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(5^{\log _{5} 8}\)

Step-by-Step Solution

Answer

The expression simplifies to 8.

Step 1: Understanding the Expression

The expression given is \(5^{\log _{5} 8}\). This is a common property in logarithms where if the base of the exponent and the base of the logarithm are the same, the expression simplifies.

Step 2: Applying the Property

According to the property of exponents and logarithms, \(a^{\log_{a} b} = b\), if the base of the exponent and the base of the logarithm are the same, which is the case here with base 5.

Step 3: Simplifying

Using the property, we can simplify \(5^{\log _{5} 8}\) directly to 8, since the base of the exponent and the base of the logarithm match (both are 5).

Key Concepts

Properties of Exponents

Exponents are a way of representing repeated multiplication of a number by itself. For example, in the expression \(a^b\), "\(a\)" is the base and "\(b\)" is the exponent. This expression means "multiply \(a\) by itself \(b\) times."
Understanding this concept is crucial, as it allows us to express large numbers in a more manageable form. Here are some important properties of exponents to remember:

• 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: \((ab)^n = a^n \cdot b^n\)
• Zero Exponent: \(a^0 = 1\) (assuming \(a eq 0\))
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

These properties help simplify complex mathematical expressions where exponents are involved. In the context of logarithmic expression simplification, they pave the way for understanding how logs and exponents interplay in equations like the original exercise here.

Properties of Logarithms

Logarithms are essentially the inverse of exponents. When you have a logarithm of a certain number, it tells you the power to which the base number is raised to get that number. For example, \( \log_a b \) asks the question: "To what power must \(a\) be raised, to produce \(b\)?"
Here are some key properties of logarithms that help in simplifying expressions:

• Product Rule: \(\log_a(m \cdot n) = \log_a m + \log_a n\)
• Quotient Rule: \(\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n\)
• Power Rule: \(\log_a(m^n) = n \cdot \log_a m\)
• Change of Base Formula: \(\log_a b = \frac{\log_c b}{\log_c a}\)

These properties form the building blocks of logarithmic operations and transformations, as they allow us to rewrite and simplify complex logarithmic expressions. In our given problem, the specific property \(a^{\log_a b} = b\) was used for simplification, taking advantage of the unique relationship between exponents and their inverse functions, logarithms.

Base of Logarithm

The base of a logarithm is a critical component, as it defines the reference number to which the logarithmic operation is linked. For example, in the expression \(\log_5 8\), "5" is the base. The base is an essential parameter without which the operations involving logs would be undefined or significantly difficult to interpret.
It's important to understand the role of the base because it is directly linked to simplifying expressions with logarithms and exponents. For instance:

• Common Logarithms have base 10, often written simply as \(\log\) without an explicit base.
• Natural Logarithms use the base \(e\), a mathematical constant approximately equal to 2.718, denoted as \(\ln\).

When the base of the exponent matches the base of the logarithm, as in the given problem, simplification becomes straightforward. This situation, \(a^{\log_a b} = b\), uses the base to directly evaluate the expression as the phenomenon relies entirely on the consistent base throughout, enabling the direct inference that the operation results in the number inside the logarithm.