Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{b}\left(\frac{\sqrt{x} y^{3}}{z^{3}}\right)$$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression is \(\frac{1}{2}\log_{b}(x) + 3\log_{b}(y) - 3\log_{b}(z)\).

Step 1: Use the Property of the Logarithm of a Quotient

We can use the property that the logarithm of a quotient is equal to the difference of the logarithms. Hence, the given expression can be expanded as \(\log_{b}(\sqrt{x} y^{3}) - \log_{b}(z^{3})\).

Step 2: Use the Property of the Logarithm of a Product

Then, we can use the property of logarithm that states the logarithm of a product is the sum of the logarithms. Thus the expression becomes \(\log_{b}(\sqrt{x}) + \log_{b}(y^{3}) - \log_{b}(z^{3})\).

Step 3: Use the Property of the Logarithm of a Power

The power of a number inside a logarithm can be moved to the front. Thus the equation will look like this: \(\frac{1}{2}\log_{b}(x) + 3\log_{b}(y) - 3\log_{b}(z)\).