Problem 12 - Calculus and its applications

Question

Solve for \(x\). $$ \log _{x} 64=3 $$

Step-by-Step Solution

Answer

The value of \( x \) is 4.

Step 1: Understanding the Logarithmic Equation

The given equation is \( \log _{x} 64=3 \). This means that \( x \) raised to the power of 3 equals 64. In mathematical terms, it is given by \( x^3 = 64 \).

Step 2: Solving the Exponentiation Equation

We need to solve the equation \( x^3 = 64 \). By taking the cube root of both sides, we find \( x = \sqrt[3]{64} \).

Step 3: Calculating the Cube Root

The cube root of 64 is 4, since \( 4^3 = 64 \). Therefore, \( x = 4 \).

Key Concepts

Cube Root

The cube root is a special kind of root used in mathematics to simplify expressions involving exponents. When we refer to the cube root of a number, we're asking what number, when multiplied by itself three times, gives us the original number. This is symbolized as \( \sqrt[3]{x} \).
For example, \( \sqrt[3]{64} \) asks us to find a number that results in 64 when cubed. In this case, 4 is the cube root of 64 because \( 4^3 = 64 \).

• Cube roots are particularly beneficial for simplifying equations, especially when dealing with cubic expressions or when solving certain types of exponential equations.
• The notation \( \sqrt[3]{x} \) is crucial to identifying cube roots distinctly from square roots, which use the simpler symbol \( \sqrt{x} \).

Knowing how to calculate cube roots can simplify many algebraic expressions and is integral to solving cubic equations.

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number, known as the base, to the power of an exponent. It is expressed in the form \( x^n \), where \( x \) is the base and \( n \) is the exponent.
The operation means multiplying the base by itself \( n \) times. For example, \( 2^3 \) means multiplying 2 by itself three times: \( 2 \times 2 \times 2 = 8 \).

• Exponentiation is useful for simplifying expressions and is invaluable when dealing with growth patterns, such as in financial calculations or population growth.
• In logarithmic and exponential equations, understanding how to manipulate exponents is key to finding solutions.
• When an exponential equation involves unknowns, it often requires logarithmic operations for solving, as seen in our case where \( x^3 = 64 \) is solved by finding \( \log_x 64 = 3 \).

Mastering exponentiation allows for faster and more precise calculations in various fields of science and mathematics.

Solve for x

Solving for \( x \) typically involves rearranging an equation to make \( x \) the subject, helping us discover its value. When dealing with logarithmic equations, such as the one given \( \log_x 64 = 3 \), this requires rewriting the equation in an exponential form.
By converting the logarithmic equation into its equivalent exponential form, we recognize that \( x^3 = 64 \). Solving for \( x \) now means finding the cube root of 64.

• To solve for \( x \) in various contexts, it often involves a combination of applying inverse operations, re-arranging terms, and simplifying the expression.
• The processes can vary based on the equation's complexity, whether it's linear, quadratic, or logarithmic.
• For logarithmic equations like the given example, sometimes a look at both logarithmic properties and basics of exponentiation is needed to find \( x \).

Solving for \( x \) is one of the most fundamental skills in algebra, leading the way to not only solving more complex equations but also for deeper understanding of mathematical relationships.