Problem 17 - Algebra and Trigonometry

Question

\(7-18\) Evaluate the expression. $$ \log \left(\log 10^{10,000}\right) $$

Step-by-Step Solution

Answer

The value of the expression \( \log(\log 10^{10,000}) \) is 4.

Step 1: Understand the Expression

The given expression is \( \log(\log 10^{10,000}) \). We need to evaluate this logarithmic expression step by step.

Step 2: Evaluate the Inner Logarithm

The inner part of the expression is \( \log 10^{10,000} \). By the property of logarithms, \( \log 10^x = x \) when the base of the logarithm is 10. So, \( \log 10^{10,000} = 10,000 \).

Step 3: Substitute and Evaluate the Outer Logarithm

Now that we know \( \log 10^{10,000} = 10,000 \), we can substitute this into the original expression to get \( \log(10,000) \). Again using the property \( \log 10^x = x \), we have \( \log(10,000) = \log(10^4) = 4 \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential when evaluating logarithmic expressions. Logarithms are mathematical expressions used to determine the power to which a number (the base) must be raised to obtain another number. Here are some key properties:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^n) = n \cdot \log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \)

In the exercise, the Power Rule is particularly relevant. The property \( \log_{10}(10^x) = x \) states that the logarithm of a base 10 raised to a power is simply the power itself. This simplifies the evaluation of logarithmic expressions greatly. It's crucial to understand these properties, as they allow for simplifying complex logarithmic expressions quickly.

Evaluating Expressions

Evaluating logarithmic expressions involves simplifying the equation using known properties and basic principles. Let's break down the original expression: \( \log(\log 10^{10,000}) \).Start by evaluating the inner expression \( \log 10^{10,000} \). Given the base 10 logarithm property, \( \log 10^x = x \), we see:

• The **inner logarithm** becomes: \( \log 10^{10,000} = 10,000 \). Hence, simplify this to 10,000.

Next, substitute the solved inner expression into the original. Now you have \( \log(10,000) \). Again, using the base 10 property, this becomes:

• \( \log(10,000) = \log(10^4) = 4 \).

Understanding these steps allows you to break down complex logarithmic expressions into more manageable calculations, ensuring each part of the expression is properly evaluated.

Step-by-Step Solutions

Approaching logarithmic questions with a clear, step-by-step method is very useful, especially for beginners. **Step 1: Understand the Expression**
Begin by identifying and understanding the structure of the expression. In this problem, the expression is \( \log(\log 10^{10,000}) \). Recognize it consists of a logarithm operation within another logarithm.**Step 2: Simplify the Inner Logarithm**
Focus on simplifying the innermost part first. For \( \log 10^{10,000} \), apply the property \( \log_{10}(10^x) = x \) to obtain 10,000. This simplifies the nested logarithm into a single numerical value.**Step 3: Solve the Outer Logarithm**
Replace the inner part with the simplified value. Now solve \( \log(10,000) \) by again applying the property for base 10 logarithms: \( \log_{10}(10^4) = 4 \). By taking each segment of the problem one step at a time, you ensure a more straightforward evaluation process. This approach builds confidence when tackling more complicated expressions in logarithms.