Certainly! Let's solve the given logarithmic expression step-by-step.
Given expression:
[tex]\[ \log_5 \sqrt{\frac{7^2 t^3 p}{d^6 b^2}} \][/tex]
To simplify this, we'll follow these steps:
1. Simplify the content inside the square root:
[tex]\[
\sqrt{\frac{7^2 t^3 p}{d^6 b^2}}
\][/tex]
First, calculate [tex]\(7^2\)[/tex]:
[tex]\[ 7^2 = 49 \][/tex]
So, inside the square root, we have:
[tex]\[
\sqrt{\frac{49 t^3 p}{d^6 b^2}}
\][/tex]
2. Take the square root of the fraction:
[tex]\[
\sqrt{\frac{49 t^3 p}{d^6 b^2}} = \frac{\sqrt{49} \cdot \sqrt{t^3} \cdot \sqrt{p}}{\sqrt{d^6} \cdot \sqrt{b^2}}
\][/tex]
Next, simplify the square roots:
[tex]\[
\sqrt{49} = 7
\][/tex]
[tex]\[
\sqrt{t^3} = t^{3/2}
\][/tex]
[tex]\[
\sqrt{p} = p^{1/2} = \sqrt{p}
\][/tex]
[tex]\[
\sqrt{d^6} = d^3
\][/tex]
[tex]\[
\sqrt{b^2} = b
\][/tex]
Putting these simplifications back into the fraction, we get:
[tex]\[
\frac{7 \cdot t^{3/2} \cdot \sqrt{p}}{d^3 \cdot b}
\][/tex]
3. Express the logarithm:
[tex]\[
\log_5 \left( \frac{7 \cdot t^{3/2} \cdot \sqrt{p}}{d^3 b} \right)
\][/tex]
4. Simplify the logarithmic expression using logarithm properties:
[tex]\[
\log_5 \left( \frac{A}{B} \right) = \log_5(A) - \log_5(B)
\][/tex]
Where [tex]\( A = 7 \cdot t^{3/2} \cdot \sqrt{p} \)[/tex] and [tex]\( B = d^3 \cdot b \)[/tex]:
[tex]\[
\log_5 \left( 7 \cdot t^{3/2} \cdot \sqrt{p} \right) - \log_5(d^3 \cdot b)
\][/tex]
Further breakdown each term into individual logs:
[tex]\[
\log_5(7) + \log_5(t^{3/2}) + \log_5(\sqrt{p}) - \left[ \log_5(d^3) + \log_5(b) \right]
\][/tex]
Convert each term:
[tex]\[
\log_5(7) + \frac{3}{2} \log_5(t) + \frac{1}{2} \log_5(p) - \left[ 3 \log_5(d) + \log_5(b) \right]
\][/tex]
Combine all terms:
[tex]\[
\log_5(7) + \frac{3}{2} \log_5(t) + \frac{1}{2} \log_5(p) - 3 \log_5(d) - \log_5(b)
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{\log(p t^3/(b^2 d^6))}{2} + \frac{\log(7)}{\log(5)}
\][/tex]
So, the fully simplified form is:
[tex]\[
\left( \frac{ \log_5(p t^3 / (b^2 d^6))}{2} + \frac{\log(7)}{\log(5)} \right)
\][/tex]
Converting to base 5 we get:
[tex]\[
\frac{\frac{1}{2}\log_5(p t^3 / (b^2 d^6)) + \log_5(7)}
\][/tex]
Let's now verify:
[tex]\[
\frac{\log (p t^3 / b^2 d^6)}{2 \log 5} + \frac{\log 7}{\log 5 }
\][/tex]
