Problem 15 - Algebra 2 and Trigonometry

Question

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log 1,024 $$

Step-by-Step Solution

Answer

\( \log 1024 \approx 3.01 \)

Step 1: Understanding the Problem

The given problem asks us to evaluate the logarithm \( \log 1024 \) to the nearest hundredth. This means we need to find the power to which 10 must be raised to get 1024, rounding the result to two decimal places.

Step 2: Recall the Logarithm Definition

The logarithm \( \log_b a \) is defined as the power \( x \) such that \( b^x = a \). For common logarithms, the base \( b \) is 10. So, we are looking for \( x \) such that \( 10^x = 1024 \).

Step 3: Use a Calculator to Evaluate the Logarithm

Since 1024 is not a power of 10, we should use a calculator to find \( \log 1024 \) directly. Entering it into a calculator, we find that \( \log 1024 \approx 3.0103 \).

Step 4: Round the Result

The result from the calculator \( 3.0103 \) needs to be rounded to the nearest hundredth. Looking at the third decimal place (1), we see it is less than 5, so we round down to \( 3.01 \).

Key Concepts

Common Logarithms

When we talk about common logarithms, we're dealing with logarithms that have a specific base: 10. This is why they are often denoted as simply \( \log \) without a subscript, as the base 10 is understood implicitly. Common logarithms are particularly useful in sciences and engineering because they simplify the process of calculating powers and roots of 10. These operations are common in those fields due to the nature of powers of ten being related to the decimal system, which aligns perfectly with our common counting method.

• For example, \( \log_{10} 100 = 2 \) because 10 raised to the power of 2 equals 100.
• Common logarithms help solve equations where the unknown variable is the exponent, allowing easier manipulation and solving of such equations.

Understanding common logarithms and their properties is essential for interpreting scientific data or solving mathematical problems involving exponential growth or decay.

Logarithm Definition

Logarithms may seem mysterious at first, but they are simply the reverse operation of exponentiation. A logarithm answers the question, "To what power must a specific base be raised to yield a given number?" For example, in the expression \( \log_b(a) = x \), \( x \) is the power to which the base \( b \) must be raised to get \( a \). This concept is crucial in understanding many math and science principles.

• If \( 10^x = 1024 \), then \( \log_{10}(1024) = x \).
• Every base must be positive and not equal to 1, ensuring the logarithm function is well-defined.
• Logarithms greatly aid in solving equations, especially when the unknown is an exponent, transforming multiplication into addition.

Grasping the logarithm definition allows you to transition more smoothly between exponential and linear thinking, a helpful skill in various real-world applications.

Rounding Decimals

Rounding decimals is a necessary step when precision limitations or readability come into play. It involves trimming extra decimal places to a specified digit limit, while adjusting the last remaining digit appropriately. When rounding to the nearest hundredth, you look at the digit in the third place.

• If the digit is 5 or greater, you round up the hundredth place digit by 1.
• If the digit is less than 5, you leave the hundredth place digit unchanged.

In our solution, we calculated \( \log_{10}(1024) \approx 3.0103 \). Since the third digit is 1, we round down, resulting in \( 3.01 \). Rounding is not just a numerical formality; it ensures that results are usable and meaningful, particularly in scientific, financial, and practical fields where exact precision may not be necessary or possible.