Question

The value of \(\log \sqrt{2 \sqrt{2 \sqrt{2 \ldots \ldots \infty \text { times }}}}+\log \sqrt{3 \sqrt{3 \sqrt{3 \ldots \ldots \infty \text { times }}}}\) is (1) 1 (2) 2 (3) \(\log 5\) (4) \(\log 6\)

Short answer

Answer: (4) \(\log 6\)

Step-by-step solution

Simplify nested square roots

Let's denote the nested square roots as: \(x = \sqrt{2 \sqrt{2 \sqrt{2 \ldots \infty \text { times }}}}\) \(y = \sqrt{3 \sqrt{3 \sqrt{3 \ldots \infty \text { times }}}}\) Now, for both \(x\) and \(y\), let's square each expression and analyze their structure: \(x^2 = 2\sqrt{2 \sqrt{2 \sqrt{2 \ldots \infty \text { times }}}}=2x\) \(y^2 = 3\sqrt{3 \sqrt{3 \sqrt{3 \ldots \infty \text { times }}}}=3y\) Solve each equation for \(x\) and \(y\): \(x = \frac{2}{\sqrt{2}}\) \(y = \frac{3}{\sqrt{3}}\) Now, we can write the given expression as: \(\log x + \log y = \log \left(\frac{2}{\sqrt{2}} \right) + \log \left(\frac{3}{\sqrt{3}}\right)\)

Use properties of logarithms to simplify the expression

Using the properties of logarithms, we can simplify the expression as follows: \(\log \left(\frac{2}{\sqrt{2}} \right) + \log \left(\frac{3}{\sqrt{3}}\right) = \log \frac{2 \cdot 3}{\sqrt{2} \cdot \sqrt{3}} = \log \frac{6}{\sqrt{6}}\) Now, we know that \(\log a + \log b = \log(ab)\), so we can rewrite the expression as: \(\log \left(\frac{6}{\sqrt{6}}\right) = \log \left(\frac{6}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}\right) = \log \left(\frac{6 \cdot \sqrt{6}}{6}\right) = \log \left(\sqrt{6}\right)^2\)

Find the value of the expression

Since \(\log a^b = b \log a\), the expression simplifies to: \(2 \log \sqrt{6} = \log 6\) Therefore, the value of the given expression is \(\log 6\). The correct option is: (4) \(\log 6\)

Key concepts

Nested Square Roots

When dealing with nested square roots, like in our exercise, we repeatedly apply the square root operation, seemingly ad infinitum. The central challenge is to simplify these expressions into something more manageable.
To do this, consider a nested expression such as \( x = \sqrt{2 \sqrt{2 \sqrt{2 \ldots \infty \text{ times }}}} \). The trick here is recognizing a pattern within the infinite structure, a way to convert infinitely nested roots into a finite expression.

By squaring both sides, we eliminate one layer of square roots: \( x^2 = 2x \). Solving for \( x \), we find \( x = \frac{2}{\sqrt{2}} \), simplifying our problem considerably. This process highlights the power of recognizing repetitive patterns in infinity, rendering them finite in practical calculation.

Properties of Logarithms

Logarithms possess unique properties that allow for simplification of otherwise complex expressions. They help in changing multiplication into addition, division into subtraction, and powers into products.

For example, if you are given the expression \( \log a + \log b \), these can be combined to form \( \log(ab) \). This property was crucial in our exercise to simplify the original logarithmic expression by converting it from a sum to a single \( \log \) term.
Another important aspect is converting radicals into exponential notation. With \( \log \left( \sqrt{a} \right) \), it can be rewritten as \( \frac{1}{2} \log a \), showcasing how logarithms transform and simplify square roots.

Simplification Techniques

Simplification techniques are essential when handling complex mathematical expressions or equations. They involve reducing expressions to their simplest form, making them easier to interpret and solve.

In our problem, simplifying was executed by breaking down nested roots and using logarithmic identities. The key steps involved squaring the nested roots to simplify them, followed by utilizing the property \( \log a + \log b = \log(ab) \) to further break down the expression. This made the calculation of \( \log 6 \) apparent.
These simplification strategies not only streamline problem-solving but also enhance comprehension of complex mathematical concepts, making them accessible and less daunting.