Problem 28 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ 5 \log _{8} 8 $$

Step-by-Step Solution

Answer

The value of \(5 \log_{8} 8\) is 5.

Step 1: Understand the Logarithmic Expression

The given expression is \(5 \log_{8} 8\). Here, \(\log_{8} 8\) represents the logarithm of 8 with base 8, and it is multiplied by 5. Our task is to evaluate this expression.

Step 2: Apply the Property of Logarithms

Recall the property of logarithms: for any positive number \(b eq 1\), \(\log_{b} b = 1\). This means that the logarithm of a number with its own base is 1. Therefore, \(\log_{8} 8 = 1\).

Step 3: Multiply by 5

Now replace \(\log_{8} 8\) with 1 in the original expression: \[5 \times \log_{8} 8 = 5 \times 1\]This simplifies to 5.

Key Concepts

Properties of Logarithms

Logarithms have some special properties that make them easier to work with. One of these properties is the identity \( \log_{b} b = 1 \), which states that the logarithm of a number with its own base is always 1. For example, \( \log_{8} 8 \) simplifies to 1 because 8 raised to the power of 1 results in 8.
Another important property is the product property: \( \log_{b}(MN) = \log_{b} M + \log_{b} N \). This means that the log of a product is the sum of the logs. We also have the quotient property: \( \log_{b}(\frac{M}{N}) = \log_{b} M - \log_{b} N \), which shows that the log of a quotient is the difference of the logs.
The power property states: \( \log_{b}(M^n) = n \times \log_{b} M \), allowing us to bring down the exponent as a coefficient. Understanding these properties enables us to manipulate and solve logarithmic expressions more easily.

Evaluating Logarithmic Expressions

When tasked with evaluating logarithmic expressions, the goal is to simplify the expression as far as possible. Let's look at the example from the exercise, \( 5 \log_{8} 8 \).
The first step is to recognize any straightforward applications of the properties of logarithms. Here, we use the identity property, \( \log_{b} b = 1 \), because the base (8) and the argument (8) are identical. This transforms \( \log_{8} 8 \) into 1.
Now that we have simplified the logarithm itself, we can move to the arithmetic step: multiplying the result by 5, as indicated in \( 5 \times \log_{8} 8 = 5 \times 1 = 5 \).
This approach can be used for evaluating other exponential expressions, ensuring you understand which properties simplify the problem efficiently, making the calculations more manageable.

Base of Logarithms

Understanding the base of a logarithm is crucial, as it defines what number the logarithm considers as a unit for measuring. In expressions like \( \log_{b} M \), the base \(b\) is the number to which the input, M, is compared in terms of powers.
A common example is the natural logarithm, where the base is \( e \approx 2.718 \), a mathematical constant. When evaluating log expressions such as \( \log_{10} M \), the base 10 is used, often referred to as a common logarithm.
When the base and argument are the same, like in \( \log_{8} 8 \), the expression simplifies immediately due to the property that \( \log_{b} b = 1 \).
Having a solid grasp of bases helps in both understanding their simplicity and manipulating complex logarithmic expressions efficiently. Always check the base first: when it's the same as the number inside the log, you have an easy path forward!