Question

use properties of logarithms to expand the logarithmic expression as much as possible. where possible, evaluate logarithmic expressions without using a calculator.
log (1000x)
log (1000x) = \square

Explanation:

Step1: Recall log product rule

The product rule of logarithms states that $\log(ab) = \log(a) + \log(b)$ for any positive numbers $a$ and $b$. Here, we have $\log(1000x)$, so we can apply this rule to split the logarithm of the product into the sum of logarithms. So, $\log(1000x)=\log(1000)+\log(x)$.

Step2: Evaluate $\log(1000)$

Assuming the logarithm is base 10 (since it's a common logarithm, usually denoted as $\log$ without a base), we know that $10^3 = 1000$. By the definition of a logarithm, if $y = \log_b(a)$, then $b^y=a$. For base 10, $\log(10^n)=n$ because $10^{\log(10^n)} = 10^n$. So, $\log(1000)=\log(10^3)=3$.

Step3: Combine the results

Substituting the value of $\log(1000)$ back into the expression from Step 1, we get $\log(1000x)=3 + \log(x)$.

Answer:

$3 + \log(x)$