Find each value. If applicable, give an approximation to four decimal places. log 0.1

Verified step by step guidance

1. Recall that the logarithm function \( \log_b a \) answers the question: "To what power must the base \( b \) be raised, to get \( a \)?" Here, the base is assumed to be 10 since it is not specified (common logarithm).
2. Rewrite the problem as \( \log_{10} 0.1 \), which means we want to find the exponent \( x \) such that \( 10^x = 0.1 \).
3. Express 0.1 as a power of 10: \( 0.1 = \frac{1}{10} = 10^{-1} \).
4. Set the equation \( 10^x = 10^{-1} \) and use the property that if \( b^x = b^y \), then \( x = y \).
5. Conclude that \( x = -1 \), so \( \log 0.1 = -1 \). If an approximation is requested, it is exactly \( -1.0000 \).

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log base 10 of 0.1 asks, '10 raised to what power equals 0.1?' Understanding this definition is essential to evaluate logarithmic expressions.

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. Knowing that log 10 = 1 and log 1 = 0 helps in estimating and calculating values like log 0.1.

Approximating Logarithmic Values

When the logarithm of a number is not an integer, it can be approximated using calculators or logarithm tables. Providing answers to four decimal places ensures precision. For example, log 0.1 equals -1 exactly, but other values may require approximation.