Problem 36 - Precalculus

Question

Evaluate the expression. \(\log \left(\sqrt[9]{10^{11}}\right)\)

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{11}{9} \).

Step 1: Understand the Expression

The problem asks us to evaluate \( \log \left( \sqrt[9]{10^{11}} \right) \). We need to simplify the expression inside the logarithm first.

Step 2: Simplifying the Radicand

The expression inside the logarithm is \( \sqrt[9]{10^{11}} \). This can also be written as \( (10^{11})^{1/9} \) by understanding that the 9th root of a number is equivalent to raising it to the power of \( 1/9 \).

Step 3: Use the Exponent Rule

Apply the exponent rule for logarithms, which states that \( \log(a^b) = b \cdot \log(a) \). Here, \( a = 10^{11} \) and \( b = 1/9 \). Therefore, \[ \log \left( (10^{11})^{1/9} \right) = \frac{1}{9} \cdot \log(10^{11}) \].

Step 4: Simplify Further Using Base 10 Logarithm

Recall that \( \log(10^n) \) simplifies to \( n \) for logarithms with base 10. This means \( \log(10^{11}) = 11 \). Substitute back into the expression: \[ \frac{1}{9} \cdot 11. \]

Step 5: Calculate the Final Value

Now, perform the multiplication: \[ \frac{1}{9} \times 11 = \frac{11}{9}. \] So, the final answer is \( \frac{11}{9} \).

Key Concepts

Radicals and Fractional Exponents

The exercise begins by introducing the concept of radicals, specifically a ninth root. A radical such as \( \sqrt[9]{10^{11}} \) can be complex at first glance, but recognizing its equivalent form as a fractional exponent simplifies the problem considerably. For any exponential expression of the form \( \sqrt[n]{a} \), it can alternatively be represented as \( a^{1/n} \). This transformation is quite powerful, enabling easier manipulation in calculations. In our example, we change \( \sqrt[9]{10^{11}} \) into \( (10^{11})^{1/9} \). This move from radical notation to fractional exponents is grounded in math's consistent rules, and it allows us to tackle the problem systematically.

Properties of Logarithms

Logarithms, by their very definition, simplify many complex operations. Utilizing properties of logarithms, such as the exponent rule, can be a game changer. The exponent rule \( \log(a^b) = b \cdot \log(a) \) translates exponential expressions into manageable forms. Consider \( \log((10^{11})^{1/9}) \): by applying this rule, we transform it into \( \frac{1}{9} \cdot \log(10^{11}) \). This showcases how logarithms help reduce high exponents into simpler arithmetic tasks. Such logarithmic properties are not just useful here but also vital in various other domains like data science and engineering, providing a bridge between complex models and their simpler representations.

Base 10 Logarithms

Among the logarithm systems, base 10, or the common logarithm, is often the most intuitive. Understanding that \( \log(10^n) = n \) when using base 10 unlocks the potential to simplify expressions quickly. In this case, since we're dealing with \( \log(10^{11}) \), the answer simplifies directly to 11. Such properties make base 10 logarithms a favored tool in mathematics and sciences for dealing with powers of ten. This step demonstrates not only the elegance of logarithms but also their practical efficiency when handling high powers, making complex computations straightforward and precise. Base 10 logarithms serve as a key component when solving problems involving exponential growth or decay in sciences and finance.