Problem 125 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln \sqrt{2}=\frac{\ln 2}{2} $$

Step-by-Step Solution

Answer

The statement is true.

Step 1: Write the square root as a power

When you write the square root of 2, this is the same as \(\2^{1/2}\). Therefore, the equation becomes \(\ln 2^{1/2} = \frac{\ln 2}{2}\).

Step 2: Apply the logarithmic property \(\ln(m^n) = n * ln(m)\)

Applying this property to \(\ln 2^{1/2}\) results in \(1/2 * \ln 2\). Therefore, the equation becomes \(1/2 * ln 2 = \frac{\ln 2}{2}\).

Step 3: Simplify the equation

Simplify the left hand side of the equation by distributing the multiplication. This simplifies the equation to \(\ln 2/2 = \ln 2/2\). Both sides of the equation are equal implying that the original statement is true.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a fundamental concept in mathematics, particularly within algebra and calculus. It is the logarithm to the base e, where \(e\) is an irrational constant approximately equal to 2.71828. This special base is natural in the sense that it arises naturally in mathematics and is the base rate of growth shared by all continually growing processes.

For instance, when we speak of compound interest that is continuously compounded, the mathematics involved would use the natural logarithm. Understanding the natural logarithm is crucial, as it helps in solving complex problems in calculus, such as finding derivatives and integrals of logarithmic functions, and aids in various scientific computations.

Furthermore, in the context of our exercise, the equation \(\ln \sqrt{2}=\frac{\ln 2}{2}\) can be understood by recognizing that the square root of a number is the same as raising that number to the power of 1/2—a concept which seamlessly integrates with the properties of natural logarithms.

Exponentiation and Logarithms

The interplay between exponentiation and logarithms is a cornerstone of algebra. Exponentiation involves raising a number to a power. For example, \(2^3\) represents 2 raised to the power of 3. Logarithms are the inverse of exponentiation. They answer the question: 'To what power must I raise a given base to get another number?' For the number \(2^3\), its logarithm base 2 would be 3.

Exponential and logarithmic functions are inverses of each other. The equation \(y = b^x\) and \(x = \log_b(y)\) are two different ways of expressing the same relationship. In the specific context of natural logarithms, if you have \(e^x\), taking \(\ln(e^x)\) will return you the power \(x\). This concept is used in our exercise where the natural logarithm helps to transform the power of \(2^{1/2}\) to a multiplication which is easier to work with.

Solving Logarithmic Equations

When it comes to solving logarithmic equations, understanding the properties of logarithms is essential. These properties include the product rule, the quotient rule, the power rule, and the change of base rule. By using these properties, one can simplify and solve logarithmic equations.

In the exercise we have, the application of the power rule \(\ln(m^n) = n \cdot \ln(m)\) is the key to verify the statement. This rule simplifies the process of dealing with logarithmic expressions involving exponents by allowing us to move the exponent to the front of the logarithm as a coefficient.

This property makes it possible to take an expression such as \(\ln 2^{1/2}\) and rewrite it as \(\frac{1}{2} \cdot \ln 2\), subsequently showing that the initial equation holds true. By mastering these properties, one can tackle otherwise intimidating logarithmic equations with confidence and ease, leading to a deeper understanding and greater mathematical prowess.