In Exercises 1–40, 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 logarithmic property that states: $\log \left(\mathrm{ab}\right)=\log \left(a\right)+\log \left(b\right)$. This means the log of a product can be written as the sum of the logs.
2. Apply this property to the expression $\log \left(1000x\right)$, which gives $\log \left(1000\right)+\log \left(x\right)$.
3. Recognize that 1000 is a power of 10, specifically $1000\; =\; 10^3$. Use the logarithmic power rule: $\log \left({10}^3\right)=\; 3\log \left(10\right)$.
4. Since $\log \left(10\right)=\; 1$ (log base 10 of 10 is 1), simplify $3\log \left(10\right)$ to 3.
5. Combine all parts to write the fully expanded expression as $3\; +\log \left(x\right)$.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. The product rule states that log(ab) = log(a) + log(b), allowing the expansion of logarithms of products into sums. These properties simplify complex logarithmic expressions and are essential for expansion.

Logarithm of Powers of 10

Logarithms with base 10 of powers of 10 are straightforward to evaluate: log(10^n) = n. For example, log(1000) equals log(10^3), which simplifies to 3. Recognizing powers of 10 helps in evaluating parts of expressions without a calculator.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves rewriting them using logarithm properties to separate terms. For example, log(1000x) can be expanded to log(1000) + log(x). This process makes it easier to simplify or evaluate each part individually.