Find each value. If applicable, give an approximation to four decimal places. log 10¹²

Verified step by step guidance

1. Recognize that the expression is \( \log 10^{12} \), which means the logarithm of \( 10^{12} \) with base 10 (common logarithm).
2. Recall the logarithm power rule: \( \log_b (a^c) = c \log_b a \). This allows us to bring the exponent down as a multiplier.
3. Apply the power rule to rewrite the expression as \( 12 \times \log 10 \).
4. Since \( \log 10 \) with base 10 is 1 (because 10 is the base of the logarithm), simplify the expression to \( 12 \times 1 \).
5. Conclude that the value of \( \log 10^{12} \) is 12. If an approximation is requested, note that this is an exact integer value.

Key Concepts

Logarithm Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to evaluate logarithmic expressions.

Logarithm of a Power

The logarithm of a number raised to a power can be simplified using the rule log_b(a^n) = n * log_b(a). This property allows us to move the exponent in front of the logarithm, making calculations easier.

Common Logarithm (Base 10)

When the base of a logarithm is 10, it is called a common logarithm and often written simply as log. For example, log(10^12) means log base 10 of 10 raised to the 12th power, which simplifies directly to 12.