Question

The value of \(\log _{5} \sqrt{5 \sqrt{5 \sqrt{5 \ldots \ldots \infty}}}+\log \left(\frac{1}{2}+\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3}+\ldots . \infty\right)\) is (1) 1 (2) 25 (3) 10 (4) 20

Short answer

Answer: The value of the given expression is 1.

Step-by-step solution

Solve the first logarithmic expression

The first expression is \(\log _{5} \sqrt{5 \sqrt{5 \sqrt{5 \ldots \ldots \infty}}}\). Let this expression be equal to x. Then we have: \(x = \log_5 \sqrt {5 \sqrt{5 \sqrt{5 \ldots \ldots \infty}}}\) Since the expression inside the square root repeats itself, we can rewrite it as: \(x = \log_5 \sqrt {5x}\)

Solve for x

To solve for x, we first remove the logarithm by raising both sides as powers of 5: \(5^x = \sqrt{5x}\) Square both sides to get rid of the square root: \((5^x)^2 = 5x\) Simplify: \(5^{2x} = 5x\) Divide both sides by 5x to isolate the exponential term: \(5^{2x - 1} = x\) Now we can either try options given in the exercise or solve it by trial and error. In this case, we can try the options: (1) If x = 1, LHS = \(5^0 = 1\) and RHS = 1; they are equal (2) If x = 25, LHS = \(5^{49} > 25\); they are not equal (3) If x = 10, LHS = \(5^{19} > 10\); they are not equal (4) If x = 20, LHS = \(5^{39} > 20\); they are not equal Since x = 1 matches both sides, the value of the first expression is 1.

Solve the second logarithmic expression

The second expression is \(\log\left(\frac{1}{2}+\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3}+\ldots . \infty\right)\). This is an infinite geometric progression with the first term \(a = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\). The sum of an infinite geometric progression is given by: \(S = \frac{a}{1-r}\) Substitute the values: \(S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1\) Now find the logarithm: \(log(1) = 0\) The value of the second expression is 0.

Find the sum of the values of the two expressions

Now we have the values for both expressions: 1st expression: 1 2nd expression: 0 Adding the values: 1 + 0 = 1 Therefore, the value of the given expression is 1. The correct answer is option (1).

Key concepts

Infinite Geometric Progression

An infinite geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. A key characteristic of an infinite geometric progression is that it continues indefinitely. The sum of this type of series can only be found when the common ratio is between -1 and 1.

To find the sum of an infinite geometric progression, use the formula:
\[ S = \frac{a}{1-r} \]where:

• \(S\) is the sum of the series.
• \(a\) is the first term.
• \(r\) is the common ratio.

In our case, the series is:\[\frac{1}{2} + \left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^3 + \ldots \infty\]Here, the first term \(a = \frac{1}{2}\) and common ratio \(r = \frac{1}{2}\). Plugging these into the formula, the series sum is 1.

This concept of summation for infinite geometric progression is crucial when dealing with similar sequences.

Sum of Logarithms

A logarithm is the inverse operation to exponentiation, and summing logarithms can simplify complex multiplication into addition. Understanding how to sum logarithms is important when dealing with logarithmic expressions or equations.

The sum of logarithms can often be handled using properties of logarithms, such as the product rule:

• \(\log_b (x) + \log_b (y) = \log_b (xy)\)

This property allows you to condense multiple logs into a single log with a multiplied argument.
In our problem, when the expression inside the logarithm is parts of an infinite geometric progression, solving for the sum first becomes crucial. Once simplified to a number with these principles, finding the logarithm is straightforward and results in simpler expressions.

Properties of Logarithms

The properties of logarithms are essential tools for simplifying and solving logarithmic expressions. They help in transforming complex expressions into manageable ones. Here are a few key properties:

• Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
• Power Rule: \(\log_b(x^y) = y \cdot \log_b(x)\)

In the exercise, simplifying the expression \(\log_5 \sqrt{5 \sqrt{5 \sqrt{5 \ldots \infty}}}\) involved recognizing and applying these properties correctly. The challenge was to express the repeating form inside the square root as a simpler expression. Using these properties allowed for breaking down the complexity into simpler and solvable parts.

Mastering these properties gives a strong foundation for tackling more advanced problems.

Exponential Equations

Exponential equations are equations where the variables appear as exponents. They often require unique approaches to solve. Let's break down the general approach to solving exponential equations:

• Equal Base Strategy: If possible, express every term as a power of the same base and then equate the exponents.
• Logarithmic Approach: Apply a logarithm to both sides to bring down the exponent, switching from an exponential form to a linear one which is easier to solve.

In solving \(5^x = \sqrt{5x}\), we first turned both sides into the same base, allowing us to manipulate the exponents directly. By squaring and removing the square root, we simplified the problem into a direct solution.

Understanding the step-by-step conversion between exponential and logarithmic forms is crucial. It helps by providing a systematic method to find the value of unknown variables in exponential equations.