Problem 17 - Precalculus Mathematics for Calculus

Question

Evaluate the expression. $$\log \left(\log 10^{10,000}\right)$$

Step-by-Step Solution

Answer

4

Step 1: Evaluate the Inner Logarithm

Start with the expression inside the logarithm: \( \log 10^{10,000} \). Using the property of logarithms, \( \log 10^x = x \cdot \log 10 \), and considering \( \log 10 = 1 \), the expression simplifies to \( 10,000 \cdot 1 = 10,000 \). So, \( \log 10^{10,000} = 10,000 \).

Step 2: Substitute the Inner Logarithm Result

Now that we have \( \log 10^{10,000} = 10,000 \), substitute this back into the original expression. The expression now is \( \log 10,000 \).

Step 3: Evaluate the Outer Logarithm

To evaluate \( \log 10,000 \), remember that this is the same as \( \log 10^4 \) because \( 10,000 = 10^4 \). Applying the logarithm property, \( \log 10^x = x \cdot \log 10 \), we get \( 4 \cdot \log 10 \). Since \( \log 10 = 1 \), the expression simplifies to \( 4 \cdot 1 = 4 \).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make them invaluable in mathematics, particularly for simplifying expressions. These properties allow us to evaluate complex logarithmic expressions easily. Let's explore some of the main properties:

• Product Property: This states that \( \log_b(MN) = \log_bM + \log_bN \). It implies that the logarithm of a product is the sum of the logarithms.
• Quotient Property: This tells us that \( \log_b\left(\frac{M}{N}\right) = \log_bM - \log_bN \). The logarithm of a quotient is the difference of the logs.
• Power Property: It affirms \( \log_b(M^n) = n\cdot\log_bM \). When a logarithm has a power, this property helps bring the exponent in front as a multiplier.
• Change of Base Formula: This property, \( \log_bM = \frac{\log_kM}{\log_kb} \), is used when you need to change the base of a logarithm for easier computation.

Understanding these properties is crucial when working with nested or complex logarithms. They are tools that allow us to break down complicated expressions into simpler terms, as seen in our exercise with \( \log 10^{10,000} \).

Evaluating Logarithms

Evaluating logarithms is all about simplifying expressions using their properties. The goal is to determine the exponent to which the base must be raised to yield the argument of the logarithm. With this understanding, you can tackle more complex logarithmic expressions.In the exercise provided, we first needed to evaluate the inner logarithm \( \log 10^{10,000} \). By applying the **Power Property**, which states that \( \log_b(M^n) = n\cdot\log_bM \), it simplifies to \( 10,000 \cdot \log 10 \). Since \( \log 10 = 1 \) in base 10 logarithms, we find that \( 10,000 \cdot 1 = 10,000 \).Once the inner logarithm has been evaluated, we substitute back into our original expression to handle the next layer of evaluation, such as \( \log 10,000 \), which further simplifies using the same principles.

Nested Logarithms

Nested logarithms appear when a logarithm is inside another logarithm. They can initially seem intimidating, but with step-by-step evaluation and use of logarithm properties, they become quite manageable.The exercise presents a typical nested logarithm: \( \log(\log 10^{10,000}) \). To evaluate this:

• First, calculate the inner part, \( \log 10^{10,000} \), using the logarithm properties, which simplifies to \( 10,000 \).
• With this result, you simplify the outer logarithm: \( \log 10,000 \). We recognize that \( 10,000 = 10^4 \), and apply the Power Property again to find that \( \log 10,000 = 4 \cdot \log 10 = 4 \).

By approaching nested logarithms this way, you systematically break down each component, making the problem easier to solve. Knowing the key properties and having a clear pathway helps you handle even the toughest looking logarithmic challenges with confidence.