Alright class, let's solve the equation \(\log _4 \sqrt{75 x+1}=2\) step by step.
Step 1: Understand the logarithmic equation.
We have the equation \(\log_4 (\sqrt{75x + 1}) = 2\).
Step 2: Convert the logarithmic equation to an exponential equation.
Recall that if \(\log_b (y) = c\), this implies \(y = b^c\). For our equation, this gives:
[tex]\[ \sqrt{75x + 1} = 4^2 \][/tex]
Step 3: Simplify the exponential equation.
Calculate \(4^2\):
[tex]\[ 4^2 = 16 \][/tex]
Thus:
[tex]\[ \sqrt{75x + 1} = 16 \][/tex]
Step 4: Remove the square root by squaring both sides.
To isolate \(75x + 1\), square both sides of the equation:
[tex]\[ (\sqrt{75x + 1})^2 = 16^2 \][/tex]
This simplifies to:
[tex]\[ 75x + 1 = 256 \][/tex]
Step 5: Solve for \(x\).
Isolate \(x\) by subtracting 1 from both sides:
[tex]\[ 75x = 256 - 1 \][/tex]
[tex]\[ 75x = 255 \][/tex]
Then, divide both sides by 75:
[tex]\[ x = \frac{255}{75} \][/tex]
Step 6: Simplify the fraction.
Reduce the fraction \(\frac{255}{75}\):
[tex]\[ x = \frac{255 \div 15}{75 \div 15} \][/tex]
[tex]\[ x = \frac{17}{5} \][/tex]
Thus, the solution to the equation \(\log _4 \sqrt{75 x+1}=2\) is:
[tex]\[ x = \frac{17}{5} \][/tex]
So, the final answer is:
[tex]\[ x = \frac{17}{5} \][/tex]
