Problem 15 - Calculus and its applications

Question

Solve for \(x\). $$ \log _{11} \sqrt{11}=x $$

Step-by-Step Solution

Answer

The value of \( x \) is \( \frac{1}{2} \).

Step 1: Understand the Logarithmic Equation

We're given the equation \( \log_{11} \sqrt{11} = x \). This implies that we need to find the power \( x \) such that when 11 is raised to this power, the result is \( \sqrt{11} \).

Step 2: Express \( \sqrt{11} \) in Exponential Form

We know that \( \sqrt{11} \) can be expressed as \( 11^{1/2} \) since taking the square root is equivalent to raising to the power of \( \frac{1}{2} \).

Step 3: Apply Logarithmic Identity

Using the property of logarithms that \( \log_b(a^c) = c \cdot \log_b(a) \), rewrite \( \log_{11}(11^{1/2}) \) as \( \frac{1}{2} \cdot \log_{11}(11) \).

Step 4: Simplify Logarithm of Base with Itself

We know that \( \log_{11}(11) = 1 \) by the definition of logarithms since a number raised to the power of 1 is itself. Therefore, \( \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Step 5: Conclude the Solution

Thus, the value of \( x \) is \( \frac{1}{2} \) since \( \log_{11}(11^{1/2}) = \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Key Concepts

Logarithmic Identities

Logarithmic identities are powerful tools that help us simplify logarithmic equations and solve for unknowns. One of the most commonly used logarithmic identities is the power rule:

• \(\log_b(a^c) = c \cdot \log_b(a)\).

This rule tells us that the logarithm of a power is the exponent times the logarithm of the base. It's particularly useful for equations involving exponents, like in our exercise.
Another important identity is the fact that the logarithm of a number with its own base is always one:

• \(\log_b(b) = 1\).

This makes sense because any number raised to the power of one is the number itself. Combining these identities allows us to break down and solve complex logarithmic equations with ease.
Understanding these identities can simplify many mathematics problems, saving time and avoiding errors.

Exponential Form

Exponential form is a way to express numbers using powers. This is crucial when working with logarithms because it directly relates to the values you’re dealing with. In the given exercise, we transformed the square root of 11 into exponential form:

• \(\sqrt{11} = 11^{1/2}\).

This is because taking a square root is equivalent to raising the base to the power of 1/2.
Understanding exponential forms is central to solving logarithmic equations, as it allows us to apply identities and simplify expressions. When you see a complicated expression under a logarithm, try to express it in exponential form. This small step can open doors to easier manipulation and quicker solutions.
By mastering the conversion between fractional exponents and roots, you’ll enhance your problem-solving skills significantly.

Properties of Logarithms

The properties of logarithms are essential tools in mathematics to simplify equations and solve for unknown variables. Some of the key properties include:

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^c) = c \cdot \log_b(M)\)
• Change of Base Formula: \(\log_b(a) = \frac{\log_k(a)}{\log_k(b)}\), where \(k\) can be any positive value.

In our exercise, the power rule allowed us to express the logarithm of \(11^{1/2}\) as a multiplication, greatly simplifying the problem.
Knowing these properties not only helps in solving equations efficiently but also deepen your understanding of how logarithms interact with other mathematical operations. They form the backbone of logarithmic manipulation and are widely used in calculus and beyond.