Problem 29 - Algebra 2

Question

Use the Change of Base Formula to evaluate each expression. Then convert it to a logarithm in base \(8 .\) $$ \log _{3} 33 $$

Step-by-Step Solution

Answer

So, the logarithmic expression \(\log_3 33\) is equivalent to approximately 0.620 when converted to a logarithm of base 8.

Step 1: Apply the Change of Base Formula

The Change of Base Formula is given by \(\log_b a = \frac{\log_c a}{\log_c b}\). We will use this formula with \(c = 10\) since it is commonly used, so the expression \(\log_3 33\) can be rewritten as \(\log_3 33 = \frac{\log_{10} 33}{\log_{10} 3}\).

Step 2: Evaluate the Equation

Now that we have the expression in terms of base 10 logarithms, it becomes straightforward to evaluate using a calculator. Doing so gives us approximately 3.137. This means \(\log_3 33 \approx 3.137\).

Step 3: Convert to a Logarithm in Base 8

Now we need to convert this result to a logarithm of base 8. We can do this by applying the Change of Base formula again: \(\log_8 x = \frac{\log_{10} x}{\log_{10} 8}\). Here, \(x = 3.137\), thus \(\log_8 3.137 = \frac{\log_{10} 3.137}{\log_{10} 8}\). Evaluating this on a calculator gives us approximately 0.620.

Key Concepts

logarithms

Logarithms are an important mathematical concept that help us solve equations involving exponential relationships. They are essentially the inverse operations of exponentiation, much like how subtraction is the inverse of addition. If you raise a number, called the base, to a certain power and get another number, a logarithm helps you find the power.The notation for logarithms is \(\log_b a\), where \(b\) is the base, and \(a\) is the result of raising \(b\) to a certain power. For example, if \(b^x = a\), then \(\log_b a = x\). Logarithms are very useful in simplifying calculations, especially when dealing with multiplication and division of exponential numbers. Key points about logarithms include:

• They convert multiplication into addition: \(\log_b (xy) = \log_b x + \log_b y\)
• They turn division into subtraction: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• Powers become products: \(\log_b (x^n) = n \cdot \log_b x\)

These properties make logarithms a powerful tool in both theoretical mathematics and practical applications like computing and engineering.

base conversion

When working with logarithms, occasionally you may encounter the need to convert them from one base to another. This is where the Change of Base Formula comes in handy. It allows you to rewrite a logarithm in a different base, making it easier to evaluate with standard calculators that typically use base 10 (common logarithms) or base \(e\) (natural logarithms).The Change of Base Formula is given by:\[\log_b a = \frac{\log_c a}{\log_c b}\]Where:

• \(b\) is the original base
• \(a\) is the number you're taking the logarithm of
• \(c\) is the new base

To apply this, you choose a new base \(c\), which is often \(10\) to simplify calculations. For example, if you have \(\log_3 33\), you can rewrite it as \(\frac{\log_{10} 33}{\log_{10} 3}\). This allows you to compute using base 10 log values that are readily available. Base conversion helps extend the usability of logarithmic functions across different systems and applications, ensuring broader computational capabilities.

mathematical expressions

Mathematical expressions convey various mathematical ideas using numbers, symbols, and operators. When dealing with logarithms, mathematical expressions often require simplification or conversion so they can be evaluated properly. This often involves applying mathematical operations systematically to achieve a desired form or value.For example, in the problem \(\log_3 33\), it requires rewriting and evaluating using the Change of Base Formula. By systematically applying the correct operations and formulas, we turn an otherwise difficult calculation into a more straightforward numerical task.To simplify or manipulate mathematical expressions involving logarithms:

• Identify the form and operations needed, such as addition or subtraction for logs.
• Use known logarithmic properties and theorems, such as change of base or rules of logarithms.
• Convert or rewrite expressions using algebraic techniques to facilitate calculation.

Understanding and manipulating mathematical expressions ensure precise and efficient solutions in various mathematical contexts, enhancing problem-solving skills and analytical thinking.