Problem 8 - 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. $$ \log _{9}\left(\frac{9}{x}\right) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression \(\log_{9}(9/x)\) is \(1 - \log_{9}x\).

Step 1: Expand using quotient property

Firstly, we will use the quotient property of logarithms to expand the logarithmic expression, we have \(\log_{9}(9/x) = \log_{9}9 - \log_{9}x\).

Step 2: Simplify using knowledge of logarithms

Since log base 9 of 9 is simply 1, the first part of the equation simplifies to 1 and thus our expression is now \(1 - \log_{9}x\). This is because any logarithm base b of b is always equal to 1.

Step 3: Final Simple Form

Since no further simplification is possible and the expression cannot be evaluated without a given value for x, the final expanded form of the given logarithmic expression is \(1 - \log_{9}x\).