Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(1000x)

Verified step by step guidance

1. Recall the logarithm property that states \( \log(ab) = \log a + \log b \). Use this to separate the logarithm of a product into the sum of logarithms.
2. Apply this property to \( \log(1000x) \), rewriting it as \( \log 1000 + \log x \).
3. Recognize that 1000 can be expressed as a power of 10: \( 1000 = 10^3 \).
4. Use the logarithm power rule \( \log a^b = b \log a \) to rewrite \( \log 1000 \) as \( \log 10^3 = 3 \log 10 \).
5. Since \( \log 10 = 1 \) (assuming base 10 logarithm), simplify \( 3 \log 10 \) to 3, so the expression becomes \( 3 + \log x \).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow you to rewrite logarithmic expressions in simpler or expanded forms. For example, the product rule states that log(ab) = log(a) + log(b), which is essential for expanding expressions like log(1000x).

Evaluating Logarithms of Powers of 10

Logarithms with base 10 of powers of 10 are straightforward to evaluate because log(10^n) = n. For instance, log(1000) equals log(10^3), which simplifies directly to 3. Recognizing these values helps simplify parts of logarithmic expressions without a calculator.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves breaking down complex logs into sums or differences of simpler logs using the properties of logarithms. This process makes it easier to simplify or evaluate the expression, such as rewriting log(1000x) as log(1000) + log(x).