To solve the equation \(\log_4 \sqrt{75x + 7} = 2\), let's follow a detailed step-by-step approach.
1. Understand the logarithm equation:
The equation given is:
[tex]\[
\log _4 \sqrt{75x + 7} = 2
\][/tex]
This means that the logarithm base 4 of \(\sqrt{75x + 7}\) equals 2.
2. Rewrite the logarithm equation:
We know that if \( \log_b(y) = z \), then \( b^z = y \). Using this property, we can rewrite the equation as:
[tex]\[
4^2 = \sqrt{75x + 7}
\][/tex]
3. Calculate the value of the exponent:
Simplify the right-hand side:
[tex]\[
4^2 = 16
\][/tex]
So the equation now reads:
[tex]\[
16 = \sqrt{75x + 7}
\][/tex]
4. Remove the square root by squaring both sides:
Squaring both sides of the equation will eliminate the square root:
[tex]\[
16^2 = (\sqrt{75x + 7})^2
\][/tex]
Simplifying this:
[tex]\[
256 = 75x + 7
\][/tex]
5. Solve for \(x\):
To isolate \(x\), subtract 7 from both sides:
[tex]\[
256 - 7 = 75x
\][/tex]
Simplifying this yields:
[tex]\[
249 = 75x
\][/tex]
Now, divide both sides by 75 to solve for \(x\):
[tex]\[
x = \frac{249}{75}
\][/tex]
6. Simplify the fraction:
Reduce the fraction \(\frac{249}{75}\) to its simplest form by finding the greatest common divisor of 249 and 75, which is 3:
[tex]\[
x = \frac{249 \div 3}{75 \div 3} = \frac{83}{25}
\][/tex]
Therefore, the solution to the equation \(\log_4 \sqrt{75x + 7} = 2\) is:
[tex]\[
x = \frac{83}{25}
\][/tex]
