Questions: log4 p = 3 log4 q = x If log4 pq - log4 256 = a x - b, then the values of a= and b= Show work.

Transcript text: \[ \begin{array}{l} \log _{4} p=3 \\ \log _{4} q=x \end{array} \] If $\log _{4} p q-\log _{4} 256=\boldsymbol{a} x-\boldsymbol{b}$, then the values of $\mathrm{a}=$ $\qquad$ and $\mathrm{b}=$ $\qquad$ Show work.

Solution

Solution Steps

Step 1: Express $ p $ and $ q $

From the given equations, we have: \[ \log_{4} p = 3 \implies p = 4^3 = 64 \] \[ \log_{4} q = x \implies q = 4^x \]

Step 2: Calculate $ \log_{4}(pq) $ and $ \log_{4}(256) $

Using the properties of logarithms: \[ \log_{4}(pq) = \log_{4}(64 \cdot 4^x) = \log_{4}(64) + \log_{4}(4^x) = 3 + x \] Next, we calculate $ \log_{4}(256) $: \[ \log_{4}(256) = \log_{4}(4^4) = 4 \]

Step 3: Formulate the Expression

Now, we can form the expression: \[ \log_{4}(pq) - \log_{4}(256) = (3 + x) - 4 = x - 1 \] We need to compare this with the form $ ax - b $.

Step 4: Identify $ a $ and $ b $

From the expression $ x - 1 $, we can see that: \[ a = 1 \quad \text{and} \quad b = 1 \]