Evaluate each expression without a calculator.


a. log₁₀ 1000

Verified step by step guidance

1. Identify the base of the logarithm. In this case, the base is 10, as indicated by the subscript in log₁₀.
2. Recall the definition of a logarithm: \( \log_b(a) = c \) means \( b^c = a \).
3. Apply the definition to the given expression: \( \log_{10}(1000) = c \) means \( 10^c = 1000 \).
4. Recognize that 1000 can be expressed as a power of 10: \( 1000 = 10^3 \).
5. Conclude that since \( 10^c = 10^3 \), it follows that \( c = 3 \).

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation. The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, in the expression log₁₀ 1000, we are looking for the power to which 10 must be raised to equal 1000.

Base of a Logarithm

The base of a logarithm indicates the number that is raised to a power. In log₁₀ 1000, the base is 10. Understanding the base is crucial because it determines the scale of the logarithmic function and how the values relate to one another.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. One important property is that logₐ (b * c) = logₐ b + logₐ c, which allows for the breaking down of complex logarithmic expressions. This property can be useful in evaluating logarithmic expressions without a calculator.