Simplifying Nested Logarithms with Base 7

The question asks us to find the value of the expression \({\log _7}{\rm{\;}}{\log _7}\sqrt {7\sqrt {7\sqrt 7 } }\). To solve this, we need to simplify the expression starting from the innermost part and work our way outwards.

Step 1: Simplify the Innermost Expression

The innermost expression is a nested square root: \(\sqrt {7\sqrt {7\sqrt 7 } }\).

We can rewrite the square root using exponents, where \(\sqrt{x} = x^{1/2}\).

• Start with the innermost \(\sqrt 7\): This is \(7^{1/2}\).
• Next, consider \(7\sqrt 7\): This is \(7 \cdot 7^{1/2} = 7^{1 + 1/2} = 7^{3/2}\).
• Then, take the square root of that: \(\sqrt {7^{3/2}} = (7^{3/2})^{1/2} = 7^{(3/2) \times (1/2)} = 7^{3/4}\).
• Now, consider \(7\sqrt {7\sqrt 7 }\): This is \(7 \cdot 7^{3/4} = 7^{1 + 3/4} = 7^{7/4}\).
• Finally, take the square root of the entire expression: \(\sqrt {7^{7/4}} = (7^{7/4})^{1/2} = 7^{(7/4) \times (1/2)} = 7^{7/8}\).

So, \(\sqrt {7\sqrt {7\sqrt 7 } } = 7^{7/8}\).

Step 2: Evaluate the First Logarithm (Base 7)

Now the expression becomes \({\log _7}{\rm{\;}}{\log _7}(7^{7/8})\). We evaluate the inner logarithm: \(\log_7(7^{7/8})\).

Using the logarithm property \(\log_b(b^x) = x\), we have:

\(\log_7(7^{7/8}) = \frac{7}{8}\).

Step 3: Evaluate the Second Logarithm (Base 7)

The expression is now reduced to \({\log _7}\left(\frac{7}{8}\right)\).

Using the logarithm property for division, \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\), we get:

\({\log _7}\left(\frac{7}{8}\right) = \log_7(7) - \log_7(8)\).

We know that \(\log_7(7) = 1\).

For \(\log_7(8)\), we can write \(8\) as \(2^3\). Using the logarithm property \(\log_b(x^k) = k\log_b(x)\), we get:

\(\log_7(8) = \log_7(2^3) = 3\log_7(2)\).

Substituting these values back, the expression becomes:

\(1 - 3\log_7(2)\).

Final Result

The value of \({\log _7}{\rm{\;}}{\log _7}\sqrt {7\sqrt {7\sqrt 7 } } \) is \(1 - 3\log_7(2)\).

Comparison with Options

Let's compare our calculated value with the given options:

• Option 1: \(3 \log_2 7\) (Does not match)
• Option 2: \(1 – 3 \log_2 7\) (Does not match, note the base of the logarithm)
• Option 3: \(1 – 3 \log_7 2\) (Matches our result)
• Option 4: \(\frac{7}{8}\) (This was an intermediate result, not the final answer)

The value of the given expression is \(1 - 3\log_7(2)\).

Revision Table: Key Logarithm and Exponent Properties

Additional Information on Logarithms and Exponents

Logarithms and exponents are inverse operations. Understanding their properties is crucial for simplifying complex expressions like the one in this question.

• Exponents: An expression like \(b^n\) means the base \(b\) is multiplied by itself \(n\) times. Properties of exponents help us simplify products, quotients, and powers involving exponential terms. In this problem, converting square roots to fractional exponents was the first key step.
• Logarithms: The logarithm \(\log_b(x)\) asks "To what power must the base \(b\) be raised to get \(x\)?" For example, \(\log_{10}(100) = 2\) because \(10^2 = 100\). Logarithm properties allow us to break down or combine logarithmic expressions, which is essential for solving logarithmic equations or simplifying expressions like \(\log_7(7/8)\). The base of the logarithm is very important; \(\log_7(2)\) is different from \(\log_2(7)\).
• Relationship: The definition of a logarithm is directly related to exponents: if \(y = \log_b(x)\), then \(b^y = x\). This inverse relationship is why \(\log_b(b^x) = x\) and \(b^{\log_b(x)} = x\).

Practicing problems involving nested roots and logarithms helps build confidence in applying these fundamental algebraic properties.