To solve the equation \( C = \frac{1}{6} \pi v^r \) for \( r \), follow these steps:
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Multiply both sides by 6 to eliminate the fraction:
\[ 6C = \pi v^r \]
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Divide both sides by \( \pi \):
\[ \frac{6C}{\pi} = v^r \]
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Take the logarithm of both sides. You can use any logarithm (common logarithm or natural logarithm). Here, we will use natural logarithm (ln):
\[ \ln\left(\frac{6C}{\pi}\right) = \ln(v^r) \]
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Use the property of logarithms that states \(\ln(a^b) = b \cdot \ln(a)\):
\[ \ln\left(\frac{6C}{\pi}\right) = r \ln(v) \]
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Finally, solve for \( r \):
\[ r = \frac{\ln\left(\frac{6C}{\pi}\right)}{\ln(v)} \]
So, the solution for \( r \) is:
\[ r = \frac{\ln\left(\frac{6C}{\pi}\right)}{\ln(v)} \]
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