Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Write the logarithm in terms of common logarithms.\(\log _{1 / 3} x\)

Step-by-Step Solution

Answer

The logarithm \( \log _{1 / 3} x \) in terms of common logarithms is \( - \frac{ \log_{10} x} { \log_{10} 3} \).

Step 1: Apply the Change of Base Formula

Apply the change of base formula to the given expression \( \log _{1 / 3} x \). The change of base formula states that \( \log_{b}a = \log_{k}a / \log_{k}b \) . With our given expression \( \log _{1 / 3} x \) , b is 1/3, a is x, and we're changing the base to 10 (common logarithm). So, applying the formula, we get: \[ \log_{1 / 3} x = \log_{10} x / \log_{10} (1 / 3) \]

Step 2: Simplify the Expression

In logarithm, \( \log_{10} (1/3) \) can be further written as \( - \log_{10} 3 \). This is possible because of the law of logarithm that states: \( \log_{b}(1 / a) = - \log_{b} a \) . So, insert this simplification into the previous equation and it becomes: \[ \log_{1 / 3} x =\log_{10} x / - \log_{10} 3 \]

Step 3: Final Result

Finally, divide \( \log_{10} x \) by \( - \log_{10} 3 \). So, \( \log_{1 / 3} x \) in terms of common logarithms is \( - \frac{ \log_{10} x} { \log_{10} 3} \).

Key Concepts

Change of Base Formula

The change of base formula is a useful tool in logarithms. It lets us rewrite a logarithm in terms of logarithms of any other base that we find more convenient to work with. This formula states that for any positive numbers a, b, and k (where b ≠ 1), the logarithm with base b of a, can be rewritten as:

• \[ \log_{b}a = \frac{\log_{k}a}{\log_{k}b} \]

This is particularly handy when dealing with calculators that only have buttons for base 10 (common logarithms) or base e (natural logarithms).
For example, in the exercise, to convert \( \log_{1 / 3} x \) into common logarithms, we use the change of base formula by letting both a and b become base 10 logarithms. So the formula becomes \( \log_{1 / 3} x = \frac{\log_{10} x}{\log_{10} (1 / 3)} \). This way, we can address any logarithm problem using a base that is easier to handle.

Common Logarithms

Common logarithms refer to logarithms that use base 10. They are called 'common' because base 10 is frequently used in calculations, particularly in scientific calculations. You will often see them expressed as \( \log_{10} \), but usually, the base is simply omitted, like \( \log \).
Common logarithms have some nice properties because they align with the base-10 number system we use in everyday counting and calculations. For instance, our numerical system is built on powers of 10, so a common logarithm directly indicates how many times you need to multiply 10 to achieve a certain number.

• Example: \( \log 100 = 2 \) because 10 must be multiplied by itself twice to equal 100.
• They are extremely useful in transforming exponential data into a manageable linear form.

With common logarithms, calculations become straightforward, particularly with the help of scientific calculators that can easily handle base 10.

Logarithm Properties

Logarithms have several rules or properties that make them a potent tool for simplifying and solving problems, particularly in scenarios involving exponential operations.
One fundamental property is the product property, which states:

• \( \log_{b}(xy) = \log_{b}x + \log_{b}y \)

This shows how a logarithm of a product can be split into a sum of logarithms, which simplifies multiplying large numbers.
Another important property is the quotient property:

• \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y \)

This helps transform division into subtraction, easing the complexity of division operations.
Lastly, the power property allows the exponent in a logarithm to be moved in front as a multiplier:

• \( \log_{b}(x^r) = r \cdot \log_{b}x \)

This is effective in simplifying expressions with exponents. These properties make it possible to address a wide range of mathematical and real-world problems efficiently using logarithms.