Problem 5 - College Algebra

Question

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. $$ \log (1000 x) $$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \(\log (1000 x)\) is \(3 + \log(x)\).

Step 1: Identify the properties of logarithms

The product rule of logarithms states that the logarithm of a product is the sum of the logarithms of its factors, i.e., \(\log(ab)=\log(a)+\log(b).\) The power rule states that the logarithm of a number raised to an exponent is the product of the exponent and logarithm of the number, i.e., \(\log(a^n)=n\log(a)\).

Step 2: Apply the product rule

The given expression is \(\log(1000x)\), which can be considered as \(\log(1000)\) and \(\log(x)\) multiplied together. Therefore, by applying the product rule, we can break it into \[\log(1000)+\log(x).\]

Step 3: Apply the power rule and simplify

The number 1000 can be written as \(10^3\). Substituting this into our expression and applying the power rule gives \[\log(10^3)+\log(x) = 3\log(10)+\log(x).\] As the base of our logarithm is \(10\), \(\log(10)\) is equal to \(1\). Therefore, our final expanded expression is \[3+\log(x).\]