Problem 38 - Algebra 2

Question

Use the properties of logarithms to evaluate each expression. \(2 \log _{8} 4-\frac{1}{3} \log _{8} 8\)

Step-by-Step Solution

Answer

So the evaluated expression is \(\frac{11}{3}\)

Step 1: Apply the logarithmic property

Rewrite the logarithmic expressions using the rules of logarithms. This will simplify the expression to: \(2 \cdot \log _{8} 4^2 - \frac{1}{3} \cdot \log _{8} 8^1\)

Step 2: Simplify the expression

By doing this we find that \(\log _{8} 4^2 = 2\log _{8} 4 = 2 \cdot 2 = 4\) and \(\log _{8} 8 = 1\). Inserting these expressions back we find \(4 - \frac{1}{3} \cdot 1 = 4 - \frac{1}{3}\).

Step 3: Finalise the calculation

Calculate the final result: \(4 - \frac{1}{3} = \frac{12}{3} - \frac{1}{3} = \frac{11}{3}\)

Key Concepts

Properties of Logarithms

Logarithms have special properties that make complex calculations easier, especially when it comes to breaking down or manipulating logarithmic expressions such as the one in our exercise. These key properties include the Product Rule, the Quotient Rule, and the Power Rule.

• Product Rule: This property states that the logarithm of a product is equal to the sum of the logarithms: \(\log_b(MN) = \log_b M + \log_b N\).
• Quotient Rule: This property simplifies the logarithm of a quotient to the difference of logarithms: \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\).
• Power Rule: This property allows the exponent to be brought in front as a multiplier, simplifying powers: \(\log_b(M^p) = p\cdot \log_b M\).

In the given problem, these rules are used to rewrite and simplify expressions, enabling more straightforward calculations.

Simplifying Expressions

When faced with a seemingly complex logarithmic expression, breaking it down using the properties mentioned above is crucial. Simplifying an expression means reshaping it to make it easier to work with, usually by reducing it to a simpler form or computing parts of it.
For example, in our exercise, we started with \(2 \log _{8} 4 - \frac{1}{3} \log _{8} 8\). The Power Rule was applied to simplify the first term to \(\log _{8} 4^2\) and the second term remained \(-\frac{1}{3} \cdot \log _{8} 8\).
The goal of simplifying is to make the expression ready for evaluation or further manipulation, reducing potential errors during calculation.

Evaluating Logarithmic Expressions

Evaluating means substituting and calculating necessary values given in the logarithmic expressions. Once the expressions have been simplified, such expressions can often be turned into straightforward numbers.
In our exercise, after simplifying, we have \(\log _{8} 4^2 = 2 \times 2 = 4\). Since \(4^2\) is 16, and \(8^x = 16\) implies \(x = 2\), the expression evaluates as expected. The second term, \(\log _{8} 8\), equals 1 since any logarithm of a base to itself is always 1.
Through evaluating expressions this way, we convert them into simple, usable numbers, allowing us to complete calculations like the subtraction needed in our problem.

Mathematical Calculations

Calculating the final result involves performing basic arithmetic operations with the numbers obtained from the evaluation of the expressions.
In the original problem, after substituting what we've calculated, we ended up with \(4 - \frac{1}{3}\). This required subtracting a fraction from a whole number, which involves understanding how to handle fractions in arithmetic.
By converting 4 into a fraction of the same denominator, \(\frac{12}{3}\), we easily subtracted \(\frac{1}{3}\) from it, resulting in \(\frac{11}{3}\). This final step ensured the precision of our overall calculations and confirmed the accuracy of our simplified and evaluated expressions.