Problem 413 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find out whether the following numerical expressions are defined or not. i. \(\sqrt{\log _{2} 1.4+\log _{2} 0.7}\). \\{Ans. Not defined ii. \(\sqrt{\log 15+\log 0.07}\). \\{ns. Defined iii. logloglog11. \\{Ans. Defined\\}

Step-by-Step Solution

Answer

i. The expression \(\sqrt{\log _{2} 1.4+\log _{2} 0.7}\) is defined. ii. The expression \(\sqrt{\log 15+\log 0.07}\) is defined. iii. The expression logloglog11 is defined.

Step 1: Analyze first expression

Evaluate the inside of the square root for \(\sqrt{\log _{2} 1.4+\log _{2} 0.7}\). Because \(\log _{2} 1.4 > 0\) and \(\log _{2} 0.7 > 0\), the sum of these logs will be greater than 0.

Step 2: Determining if the first expression is defined

Because the result inside the square root operation is positive, the entire expression is defined as square roots are only undefined when the argument (input) is negative.

Step 3: Analyze second expression

Evaluate the inside of the square root for \(\sqrt{\log 15+\log 0.07}\). Both expressions \(\log 15\) and \(\log 0.07\) are defined and therefore so is the sum inside the square root.

Step 4: Determining if the second expression is defined

Because both \(\log 15\) and \(\log 0.07\) are defined, as is their sum, the entire expression is defined because the argument inside the square root operation is not negative.

Step 5: Analyze third expression

Looking at the expression 'logloglog11'. The outermost log is defined if the 'loglog11' is positive. Analyze 'loglog11'. This is also defined if 'log11' is positive. 'log11' is indeed positive because log of any number greater than 1 is positive.

Step 6: Determining if the third expression is defined

Since each successive log is positive, the entire expression 'logloglog11' is defined.

Key Concepts

Understanding Square Roots in Expressions

Square roots are a fundamental part of mathematics. They represent a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 \times 3 = 9. However, when dealing with expressions, understanding square roots requires knowing what values are permissible inside the radical sign.
For the square root function to output a real number, the number under the square root, known as the radicand, must be non-negative (i.e., zero or positive). If the radicand becomes negative, the square root is not defined in real numbers.

• For expression \(\sqrt{\log _{2} 1.4+\log _{2} 0.7}\), check if the radicand (sum of logs) is non-negative.
• Similarly, for \(\sqrt{\log 15+\log 0.07}\), ensure the sum produces a non-negative result to define it.

In our exercise, both expressions eventually show positive results under the square roots, so they are defined except the one initially assumed in error.

Exploring Logarithms in Expressions

Logarithms are an essential concept that helps in dealing with multiplicative processes, such as exponential growth and decay in science and finance. The logarithm of a number is the power to which the base must be raised to produce that number. For instance, \(\log_{10} 100\) answers the question: "To what power must 10 be raised to get 100?" The answer is 2 because \(10^2 = 100\).
Common logarithms involve base 10, while natural logs use the base \(e\), which is approximately 2.718.

• When we see \(\log 15\), it means we're looking for the power that gets us to 15 using the base 10.
• Similarly, \(\log 0.07\) is defined because 0.07 is positive, and logs are typically defined for positive numbers.

Thus, within expressions, ensure each log part is positive before further evaluating, ensuring the expression is defined.

Evaluating Expressions: A Step-by-Step Guide

To evaluate expressions effectively, it's crucial to break them down into parts and ensure each segment is logical and well-defined before combining them. This comes into play when dealing with nested operations like multiple logarithms or arithmetic operations in an expression.
For example, in nested expressions such as "logloglog11" :

• Start by considering the innermost part, \(\log 11\). Since 11 is greater than 1, \(\log 11\) is positive.
• Next, consider "\(\log(\log 11)\)". As \(\log 11\) is positive, \(\log(\log 11)\) is defined.
• Finally, assess "\(\log(\log(\log 11))\)" to ensure its positivity. When each segment is positive, the overall expression is defined.

By following this step-by-step process, you ensure every part of complex expressions is coherent, allowing for correct evaluation and understanding. Developing this habit of checking sequentially prevents errors and misunderstandings in math problems.