Problem 48 - Precalculus with Limits

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. \(\log 3 z=2\)

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\log_{3} z = 2 \) is \( z = 9.000 \)

Step 1: Transform to Exponential Form

Transform the given logarithmic equation \(\log_{3} z = 2 \) to exponential form. This means expressing it in terms of base and exponent. In exponential form, a logarithm \(\log_{b} a = m \) is equivalent to \( b^m = a \). Therefore, in this case, the exponential form would be \( 3^2 = z \)

Step 2: Compute 'z'

Next, compute the value of the variable 'z' using the equation obtained in step 1: \( 3^2 = z \). This calculation simplifies to \( z = 9 \).

Step 3: Approximate the Result

Even though 'z' is already a whole number in this case, the problem requires us to provide the result to three decimal places. Thus, 'z' should be expressed as \( z = 9.000 \)

Key Concepts

Understanding Exponential Form

When dealing with logarithmic equations, a fundamental skill is transforming them into exponential form. Knowing how these two mathematical concepts are interconnected can drastically simplify solving equations.

A logarithmic statement such as \( \log_b a = m \) can be rewritten in exponential form as \( b^m = a \). Essentially, the base of the logarithm raised to the exponent (the number on the right side of the equation) equals the number inside the log.

For instance, let's consider the logarithmic equation \( \log_{3} z = 2 \). When transformed into its exponential form, it tells us that raising the base (3) to the power of 2 results in \( z \), that is, \( 3^2 = z \). This transformation is incredibly helpful as it straightforwardly gives us the equation \( z = 9 \), ready for resolution.

Solving Logarithmic Equations

Solving logarithmic equations involves a few systematic steps, predominantly revolving around their exponential relationships.

To begin, the key step is to transform the logarithmic equation into its equivalent exponential form. This transformation, as seen earlier, reduces the problem to a more recognizable form of an equation.

Subsequently, once the problem is in exponential form, the equation can be solved by performing basic arithmetic calculations.

• Identify the base of the log and transform the equation accordingly. For example, \( \log_{3} z = 2 \) becomes \( 3^2 = z \).
• Compute the necessary operations: \( 3^2 \), in this case, calculates to \( 9 \), which reveals that \( z = 9 \).

The final step is always verifying the calculation to ensure the solution meets the initial logarithmic equation conditions.

Approximating to Decimals

Approximating to decimals is a common requirement in mathematical solutions, especially when the result needs to be presented in a specific format or significant figures are important.

While the step might seem trivial when the computed result is a whole number, like \( z = 9 \), the approximation process becomes critical in other circumstances where more precise values are needed.

For example, a result expressed as \( 9 \) may need to be recorded as \( 9.000 \) to fulfill requirements for accuracy to three decimal places. This method ensures uniformity and precision in solutions, which is particularly useful in fields like science and statistics.

• Transform whole numbers by adding zero placeholders: \( 9 \) becomes \( 9.000 \).
• For non-whole numbers, standard rounding rules apply to achieve the desired decimal precision.

Adhering to these decimal approximations ensures the solution's clarity and accuracy for further applications.