Problem 35 - Algebra A Combined Function

Question

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log x=2.3 $$

Step-by-Step Solution

Answer

The exact solution is \( x = 10^{2.3} \). The four-decimal-place approximation is 199.5262.

Step 1: Convert to Exponential Form

The logarithmic equation \( \log x = 2.3 \) can be converted to an exponential form. Remember that \( \log x = y \) implies \( 10^y = x \). So, \( x = 10^{2.3} \).

Step 2: Calculate Exact Solution

The exact solution for \( x \) is given by the expression \( x = 10^{2.3} \). This is the exact value because it's expressed in terms of powers of 10.

Step 3: Approximate the Solution

To find a numerical approximation, calculate \( 10^{2.3} \) using a calculator. Raise 10 to the power of 2.3, which results in approximately 199.5262. Round this to four decimal places to get 199.5262.

Key Concepts

Exponential Form

When faced with a logarithmic equation like \( \log x = 2.3 \), understanding how to convert it into an exponential form is crucial. This conversion is rooted in the basic definition of logarithms. The logarithm tells you the power to which the base must be raised to get a certain number. In our equation, the base is 10, and the logarithm tells us that when you raise 10 to the power of 2.3, you get \( x \).
Thus, we restate the equation in exponential form as \( 10^{2.3} = x \). This step is essential because it allows us to work with the number \( x \) in terms of exponents, which is often easier to calculate with a scientific calculator or when explaining concepts of growth and decay in applications like finance or biology.

• Base: 10, which is the standard base for common logarithms.
• Exponent: 2.3, derived from the logarithmic equation.
• Result: \( x = 10^{2.3} \).

Exact Solution

Finding the exact solution of a logarithmic equation involves expressing the answer in its most precise mathematical form. Since we previously converted our logarithmic expression to exponential form as \( x = 10^{2.3} \), this expression itself is the exact solution.
It's important to understand that while \( 10^{2.3} \) might seem abstract without calculating a specific numerical value, this is often the kind of precision needed in higher-level mathematics or theoretical applications.

• Advantages of an Exact Solution:
  • Retains full mathematical precision without rounding errors.
  • Useful in symbolic computation and further theoretical analysis.
• Applications: Exact solutions are crucial in theoretical physics, engineering, and any field that requires rigorous mathematical proof.

Numerical Approximation

Numerical approximation takes the exact mathematical expression and converts it into a decimal number that is easy to understand and use practically. For the equation \( x = 10^{2.3} \), using a calculator we find that this evaluates to approximately 199.5262. Rounding this to four decimal places, we determine the approximation as 199.5262, which means our final expression is fairly precise.
This process is essential when you need a quick, easy-to-use solution, such as for everyday calculations or when a high degree of precision isn't crucial. Calculators are excellent tools for numerical approximations, ensuring our final numbers are accurate enough for practical purposes.

• Use of Calculators: Allows for easy computation of powers and logs.
• Precision: Although rounded, the four-decimal precision is often sufficient for most practical tasks.
• Application: Useful for financial calculations, physics simulations, and any other scenarios where precise modeling of reality can benefit from quick computations.