Problem 32 - Algebra 2

Question

Use the Change of Base Formula to evaluate each expression. Then convert it to a logarithm in base \(8 .\) $$ \log _{4} 1.116 $$

Step-by-Step Solution

Answer

The log base 4 of 1.116 when converted to log base 8 is approximately \(-0.69445\).

Step 1: Apply the Change of Base Formula

We will transform the given expression using the change of base formula. \(\log _{4} 1.116 = \frac{\log 1.116}{\log 4}\)

Step 2: Compute the Result

By performing the division operation in the previous step, we obtain the value in base 10, which is approximately 0.08399.

Step 3: Convert Result into Base 8

Finally, convert the resulting value to a logarithm in base 8. You can use the Change of Base formula again to convert to base 8 as follows: \(\log_{8}(0.08399) = \frac{\log10(0.08399)}{\log10(8)}\) . Once the right-side division operation is performed, the result gives the logarithm of 0.08399 in base 8.

Key Concepts

Understanding Logarithms

Logarithms might initially seem complex, but they are actually a fundamental part of mathematics that simplifies multiplication and division operations by converting them into addition and subtraction, respectively. At its core, a logarithm is the inverse operation to exponentiation, similar to how subtraction undoes addition and division undoes multiplication. A basic logarithm in the form of \(\log_b a = c\) means "the exponent \(c\) such that \(b^c = a\)." Here, \(b\) is the base, \(a\) is the argument of the logarithm, and \(c\) is the resulting value.
Understanding the properties and rules of logarithms, such as \(\log_b (xy) = \log_b x + \log_b y\), \(\log_b (\frac{x}{y}) = \log_b x - \log_b y\), and \(\log_b (x^c) = c \cdot \log_b x\), can be very helpful in solving logarithmic equations and understanding their applications in various subjects like algebra and calculus.

Base Conversion in Logarithms

Base conversion in logarithms is necessary when you want to change the base of a logarithmic expression, such as converting \(\log_4 1.116\) into a common logarithmic base like 10 or an unconventional base like 8. This process uses the Change of Base Formula, which states: \(\log_b a = \frac{\log_c a}{\log_c b}\).
Here, you initially convert the logarithm to its base 10 form (common logarithms), which is convenient as most calculators can easily evaluate these. In our original exercise, we started with \(\log_4 1.116\) and converted it to base 10 by applying the formula.
Later, adjusting this result to another base, such as base 8, involves applying the same formula again but with your desired base in the denominator: \(\log_8 0.08399 = \frac{\log_{10} 0.08399}{\log_{10} 8}\). Converting logarithmic bases can make calculations simpler or align them with the proper base system used in a specific context, like computer science or other applied fields.

Using Algebraic Expressions with Logarithms

When handling logarithms in algebra, they are often manipulated through algebraic expressions for simplification or solving purposes. Algebraic expressions provide a structured way to represent and work with numbers, including unknown variables and constants using operations such as addition, subtraction, multiplication, division, powers, and roots.
In mathematical problems, we frequently integrate logarithms within these expressions to solve for unknowns or to simplify complex problems involving exponential growth or decay. For instance, the original exercise requires using an algebraic manipulation called the Change of Base Formula to transform \(\log_4 1.116\) into an expression with a different base, showing the interplay between algebra and logarithms.
By applying algebraic skills, one can derive solutions that involve evaluating or converting logarithms, and therefore solve real-world problems or solve mathematical theorems, making understanding both concepts critical in higher mathematics. The integration of algebraic expressions seamlessly with logarithms facilitates the computation and understanding of otherwise intricate calculations.