Problem 25 - 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. $$\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)$$

Step-by-Step Solution

Answer

The expanded and evaluated logarithmic expression is : \(2 - (1/2)\log_6{(x+1)}\).

Step 1: Expand Logarithm

Firstly, use the properties of logarithms to expand the given expression. For a given base \(b\), \(\log_b(m/n) = \log_b m - \log_b n\) and conversely, \(\log_b \sqrt[n]{m} = (1/n) \log_b m\). Therefore, \(\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)\) becomes \(\log_6{36} - \log_6{\sqrt{x+1}}\).

Step 2: Decomposition of Logarithm with Radix

Having expanded the fraction, the next step involves dealing with the square root (i.e., the radical symbol) within the logarithm. \(\log_6{\sqrt{x+1}}\) decomposes as \((1/2)log_6{(x+1)}\) according to logarithmic properties.

Step 3: Evaluating the Logarithmic Expressions

Finally, number values of the logarithm can be input where possible to render the expression as precise as feasible without the necessity of a calculator. It is known that \(\log_b{b^n} = n\), so \(\log_6{36}\) or \(\log_6{6^2}\) results in 2 as 36 is 6 squared. This leads to the final expression 2 - \((1/2)log_6{(x+1)}\).