Problem 13 - Calculus and its applications

Question

Solve for \(x\). $$ \log _{3} x=5 $$

Step-by-Step Solution

Answer

\( x = 243 \)

Step 1: Understand the Logarithmic Equation

The equation is given as \( \log_{3} x = 5 \). This means that the logarithm with base 3 of \( x \) is equal to 5. In other words, \( 3^5 = x \).

Step 2: Apply the Exponential Form

Using the property of logarithms where if \( \log_{b} a = c \), then \( b^c = a \), we can rewrite the equation \( \log_{3} x = 5 \) as \( x = 3^5 \).

Step 3: Calculate the Exponentiation

Now, calculate \( 3^5 \). The calculation is as follows: \( 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \). Therefore, \( x = 243 \).

Key Concepts

Exponential Form

Exponential form is a way of writing numbers using exponents. It is particularly handy when dealing with logarithmic equations. When you have a logarithmic expression like \( \log_{b} a = c \), it can be rewritten in exponential form as \( b^c = a \). In this transformation, the base of the logarithm (\(b\)) becomes the base of the exponent (\(b\)), and the logarithm's result (\(c\)) becomes the exponent in this new equation.

This is exactly what happens when we solve \( \log_{3} x = 5 \). By converting it into its exponential form, \( x = 3^5 \), we make it easier to find the value of \(x\). This approach leverages the nifty transformation properties between logarithms and exponents, allowing us to manipulate equations more freely.

Properties of Logarithms

Understanding the properties of logarithms can simplify solving equations that involve them. Here are some important logarithm properties you might find helpful:

• Product Property: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient Property: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• Power Property: \( \log_b (M^k) = k \cdot \log_b M \)

These properties are useful for rewriting expressions and making complex logarithmic equations more manageable.
Applied correctly, these properties do not only simplify calculations, but they also provide a deeper understanding of how logarithms function logically. This understanding is crucial when transforming a logarithmic equation into something more solvable, such as its exponential form.

Solving Equations

To solve equations, especially those involving logarithms or exponents, it's important to have a clear strategy. Start with what the equation presents you and identify any mathematical properties that can simplify the process.

• Identify the problem type: Determine if it involves logarithms, exponents, or both.
• Use properties effectively: Apply relevant mathematical properties to eliminate complexities.
  
  
• Transform if needed: Change the equation's form (from logarithmic to exponential, for instance) to ease calculations.
• Perform calculations: Once simplified or transformed, solve the equation as you normally would.

For our case, transforming the logarithmic equation into exponential form allowed us to compute \( x \) directly by calculating \( 3^5 \). The process becomes methodical: understand, simplify, transform, and compute. Having a structured approach helps tackle not just this specific problem, but any logarithmic or exponential equation confidently.