Problem 62 - Algebra 2

Question

Show that each statement is true. \(\log _{7}\left[\log _{3}\left(\log _{2} 8\right)\right]=0\)

Step-by-Step Solution

Answer

The statement is true. \(\log_{7} [\log_{3} (\log_{2} 8)] = 0\).

Step 1: Understand the Innermost Logarithm

First, evaluate the innermost logarithmic expression, \(\log_{2} 8\). Since 8 is \(2^3\), \(\log_{2} 8 = 3\).

Step 2: Evaluate the Middle Logarithm

Now, move to the next layer: \(\log_{3}(3)\). Since we found that \(\log_{2} 8 = 3\), this becomes \(\log_{3}(3)\), which is 1 because any logarithm \(\log_{b}(b) = 1\).

Step 3: Evaluate the Outermost Logarithm

Finally, compute the outermost expression: \(\log_{7}(1)\). Any logarithm \(\log_{b}(1) = 0\) because any number raised to the power of 0 is 1.

Key Concepts

Understanding the Properties of Logarithms

Logarithms are fascinating mathematical functions that help us solve for unknowns in equations, particularly where exponents are involved. They have several properties that make them highly versatile. One essential property is that

• The logarithm of a number where the base and the argument are the same is 1, mathematically shown as \( \log_{b}(b) = 1 \), because any number raised to the power of 1 equals itself.


• Another important property is that the logarithm of 1 with any base is always 0: \( \log_{b}(1) = 0 \), as any number raised to the power of 0 results in 1.

These properties play a critical role in simplifying complex logarithmic expressions, allowing us to reduce them step by step.

Deciphering Nested Logarithmic Expressions

Nested logarithmic expressions can initially seem intimidating, but breaking them down simplifies the process. Imagine peeling an onion layer by layer. Let's break it down:

• Identify the innermost expression. For example, given \( \log_{7}\left[\log_{3}\left(\log_{2} 8\right)\right] \), start with \( \log_{2} 8 \), which simplifies based on the property that if a base raised to a specific power equals the number (e.g., 8 is \(2^3\)), the result is 3.


• Evaluate each following layer using known logarithmic identities, as we do with \(\log_{3}(3) \) to yield 1.


• Conclude with the outermost layer, which often reduces to simple properties, here \(\log_{7}(1) = 0\).

By addressing each layer individually, nested logarithms become highly manageable.

Efficiently Evaluating Logarithmic Equations

Solving logarithmic equations involves a series of logical steps based on the properties of logarithms. When dealing with equations like \(\log _{7}\left[\log _{3}\left(\log _{2} 8\right)\right] = 0\), we approach as follows:

• First, simplify each logarithmic term step-by-step, starting from the innermost. These are often straightforward once the base and number align nicely, like \( \log_{2} 8 = 3 \).


• Second, using properties such as \(\log_{b}(b) \text{ and } \log_{b}(1)\), continue simplifying the expression until it resolves to a basic logarithmic property.


• Finally, confirm each step to ensure accuracy in the logical progression, checking each simplification holds true until the equation is fully evaluated.

Approaching in stages ensures no detail is overlooked, leading to accurate solutions each time.