Question

C=1/6 πv^r solve for r

Answer 1

To solve the equation \( C = \frac{1}{6} \pi v^r \) for \( r \), follow these steps:

1. **Multiply both sides by 6 to eliminate the fraction:**

\[
6C = \pi v^r
\]

2. **Divide both sides by \( \pi \):**

\[
\frac{6C}{\pi} = v^r
\]

3. **Take the logarithm of both sides. You can use any logarithm (common logarithm or natural logarithm). Here, we will use natural logarithm (ln):**

\[
\ln\left(\frac{6C}{\pi}\right) = \ln(v^r)
\]

4. **Use the property of logarithms that states \(\ln(a^b) = b \cdot \ln(a)\):**

\[
\ln\left(\frac{6C}{\pi}\right) = r \ln(v)
\]

5. **Finally, solve for \( r \):**

\[
r = \frac{\ln\left(\frac{6C}{\pi}\right)}{\ln(v)}
\]

So, the solution for \( r \) is:

\[
r = \frac{\ln\left(\frac{6C}{\pi}\right)}{\ln(v)}
\]