Problem 11 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$8^{\log _{8} 5}$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Understand the Expression

The given expression is \(8^{\log_{8} 5}\), which involves an exponentiation with a logarithm.

Step 2: Recognize the Logarithmic Identity

We apply the identity \(a^{\log_{a} b} = b\) where the base \(a\) of the exponent matches the base of the logarithm.

Step 3: Apply the Identity

Following the identity, substitute \(a = 8\) and \(b = 5\) into \(a^{\log_{a} b} = b\). This simplifies \(8^{\log_{8} 5}\) to \(5\).

Key Concepts

Logarithmic Identity

When dealing with expressions involving both exponents and logarithms, such as \(8^{\log_{8} 5}\), it is crucial to understand the concept of a logarithmic identity. Logarithmic identities are like shortcuts in math that help streamline complex equations. Specifically, the identity \(a^{\log_{a} b} = b\) plays a vital role in simplifying expressions where the base of the exponent and logarithm are the same.

This identity is based on the definition of logarithms where \(\log_{a} b\) represents the exponent to which the base \(a\) must be raised to yield \(b\). So, when you have \(a\) raised to this power (i.e., \(\log_{a} b\)), you simply get \(b\). It captures the essence of reversing the logarithmic process and is immensely useful in problems involving log bases corresponding to exponential expressions. Understanding this concept thoroughly makes it easier to tackle logarithmic and exponential problems without feeling overwhelmed.

In our example, it's clear that \(8^{\log_{8} 5} = 5\). The identity removes the complication of the logarithm and reveals the number directly.

Exponentiation

Exponentiation is a mathematical operation that extends the idea of repeated multiplication. For instance, \(8^2\) means multiplying \(8 \times 8 = 64\). Here, \(8\) is the base, and \(2\) is the exponent, telling us how many times the base is used as a factor.

In the expression \(8^{\log_{8} 5}\), exponentiation combines with a logarithm, giving context to how much more powerful the exponent can become. Exponents can simplify expressions significantly, especially when coupled with logarithms through identities.

Essentially, exponentiation is central to the mathematical exploration of growth and scaling. In fields ranging from science to finance, understanding exponentiation allows you to grasp the magnitude changes exponentially. When linked to the logarithm, as seen in our expression, it efficiently computes values without needing mechanical aids like calculators.

Simplifying Expressions

Simplifying mathematical expressions is about breaking them down into their simplest form while keeping their value intact. This makes expressions easier to work with and understand.

In the expression \(8^{\log_{8} 5}\), simplifying involves recognizing that the logarithmic identity \(a^{\log_{a} b} = b\) can be applied. By doing this, we reduce a potentially complex calculation to a simple number: \(5\).

Simplifying uses various maths operations, where:

• Logarithmic identities deal with relationships between exponents and logs.
• Arithmetic simplification reduces expressions using basic math operations like addition and multiplication.

In practical terms, simplification turns difficult-to-comprehend strings of variables and numbers into straightforward values, aiding in clearer communication and quicker solutions in both academic and real-world mathematical applications.