Questions: Write the expression as a single logarithm. Express powers as factors. log6 sqrt(x) - log6 x^5 log6 sqrt(x) - log6 x^5= (Type an exact answer. Use integers or fractions for any numbers in the expression.)

$Write the expression as a single logarithm. Express powers as factors. log6 sqrt(x) - log6 x^5 log6 sqrt(x) - log6 x^5= (Type an exact answer. Use integers or fractions for any numbers in the expression.)$

Transcript text: Write the expression as a single logarithm. Express powers as factors. \[ \begin{array}{c} \log _{6} \sqrt{x}-\log _{6} x^{5} \\ \log _{6} \sqrt{x}-\log _{6} x^{5}= \end{array} \] $\square$ (Type an exact answer. Use integers or fractions for any numbers in the expression.)

Solution

Solution Steps

Step 1: Rewrite the Original Expression

We start with the expression: \[ \log_{6} \sqrt{x} - \log_{6} x^{5} \]

Step 2: Apply the Logarithmic Property

Using the property of logarithms that states $\log_{b} a - \log_{b} c = \log_{b} \left( \frac{a}{c} \right)$, we can combine the logarithms: \[ \log_{6} \sqrt{x} - \log_{6} x^{5} = \log_{6} \left( \frac{\sqrt{x}}{x^{5}} \right) \]

Step 3: Simplify the Argument

Next, we simplify the argument of the logarithm: \[ \frac{\sqrt{x}}{x^{5}} = \frac{x^{1/2}}{x^{5}} = x^{1/2 - 5} = x^{-4.5} \]

Step 4: Write the Final Expression

Thus, we can express the original logarithmic expression as: \[ \log_{6} \left( x^{-4.5} \right) \]