Solve each problem. Use a calculator to find an approximation for each logarithm. log 398.4

Verified step by step guidance

1. Recognize that the problem asks for the logarithm of 398.4, which is written as \(\log 398.4\). This typically means the common logarithm, or logarithm base 10.
2. Recall the definition of the common logarithm: \(\log x\) is the exponent to which 10 must be raised to get \(x\). So, \(\log 398.4 = y\) means \$10^y = 398.4$.
3. Use the logarithm properties to break down the number if needed. For example, you can write \(398.4\) as \(3.984 \times 10^2\), so \(\log 398.4 = \log (3.984 \times 10^2)\).
4. Apply the product rule of logarithms: \(\log (ab) = \log a + \log b\). So, \(\log (3.984 \times 10^2) = \log 3.984 + \log 10^2\).
5. Since \(\log 10^2 = 2\), the expression simplifies to \(\log 3.984 + 2\). Use a calculator to find \(\log 3.984\) and then add 2 to get the approximate value of \(\log 398.4\).

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must a base be raised to produce a given number? For example, log base 10 of 100 is 2 because 10 squared equals 100. Understanding this helps interpret what the logarithm represents.

Common Logarithms (Base 10)

Common logarithms use base 10 and are often written simply as log without a base. They are widely used in science and engineering. Calculating log 398.4 means finding the power to which 10 must be raised to get 398.4.

Using a Calculator for Logarithms

Calculators have a log function to quickly find logarithms of numbers. To approximate log 398.4, enter 398.4 and press the log button. This provides a decimal approximation, useful when exact values are difficult to compute manually.