Solve the equation \( 15^{-6p-6}-6.2=62 \).
Solve the equation by following steps:
- step0: Solve for \(p\):
  \(15^{-6p-6}-6.2=62\)
- step1: Move the constant to the right side:
  \(15^{-6p-6}=62+6.2\)
- step2: Add the numbers:
  \(15^{-6p-6}=68.2\)
- step3: Rewrite the expression:
  \(15^{-6p-6}=\frac{341}{5}\)
- step4: Take the logarithm of both sides:
  \(\log_{15}{\left(15^{-6p-6}\right)}=\log_{15}{\left(\frac{341}{5}\right)}\)
- step5: Evaluate the logarithm:
  \(-6p-6=\log_{15}{\left(\frac{341}{5}\right)}\)
- step6: Move the constant to the right side:
  \(-6p=\log_{15}{\left(\frac{341}{5}\right)}+6\)
- step7: Change the signs:
  \(6p=-\log_{15}{\left(\frac{341}{5}\right)}-6\)
- step8: Divide both sides:
  \(\frac{6p}{6}=\frac{-\log_{15}{\left(\frac{341}{5}\right)}-6}{6}\)
- step9: Divide the numbers:
  \(p=-\frac{\log_{15}{\left(\frac{341}{5}\right)}+6}{6}\)
- step10: Simplify:
  \(p=\frac{-\log_{15}{\left(341\right)}+\log_{15}{\left(5\right)}-6}{6}\)
- step11: Simplify:
  \(p=\frac{\log_{15}{\left(\frac{5}{341}\right)}-6}{6}\)
- step12: Evaluate:
  \(p=\frac{\log_{15}{\left(5\right)}-\log_{15}{\left(341\right)}-6}{6}\)
The solution to the equation \(15^{-6p-6}-6.2=62\) is \(p=\frac{\log_{15}(5)-\log_{15}(341)-6}{6}\).

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Question

25) \( 15^{-6 p-6}-6.2=62 \)

Ask by Cummings Martin. in the United States

Feb 02,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

\( p = \frac{\log_{15}(5) - \log_{15}(341) - 6}{6} \)

Solution

Solve the equation \( 15^{-6p-6}-6.2=62 \). Solve the equation by following steps: - step0: Solve for \(p\): \(15^{-6p-6}-6.2=62\) - step1: Move the constant to the right side: \(15^{-6p-6}=62+6.2\) - step2: Add the numbers: \(15^{-6p-6}=68.2\) - step3: Rewrite the expression: \(15^{-6p-6}=\frac{341}{5}\) - step4: Take the logarithm of both sides: \(\log_{15}{\left(15^{-6p-6}\right)}=\log_{15}{\left(\frac{341}{5}\right)}\) - step5: Evaluate the logarithm: \(-6p-6=\log_{15}{\left(\frac{341}{5}\right)}\) - step6: Move the constant to the right side: \(-6p=\log_{15}{\left(\frac{341}{5}\right)}+6\) - step7: Change the signs: \(6p=-\log_{15}{\left(\frac{341}{5}\right)}-6\) - step8: Divide both sides: \(\frac{6p}{6}=\frac{-\log_{15}{\left(\frac{341}{5}\right)}-6}{6}\) - step9: Divide the numbers: \(p=-\frac{\log_{15}{\left(\frac{341}{5}\right)}+6}{6}\) - step10: Simplify: \(p=\frac{-\log_{15}{\left(341\right)}+\log_{15}{\left(5\right)}-6}{6}\) - step11: Simplify: \(p=\frac{\log_{15}{\left(\frac{5}{341}\right)}-6}{6}\) - step12: Evaluate: \(p=\frac{\log_{15}{\left(5\right)}-\log_{15}{\left(341\right)}-6}{6}\) The solution to the equation \(15^{-6p-6}-6.2=62\) is \(p=\frac{\log_{15}(5)-\log_{15}(341)-6}{6}\).

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Simplify this solution

Mind Expander

To solve the equation \( 15^{-6p-6} - 6.2 = 62 \), we first isolate the term involving the exponent. Adding \( 6.2 \) to both sides gives: \[ 15^{-6p-6} = 62 + 6.2 = 68.2 \] Now we can take the logarithm of both sides. Using base 10 or natural logarithm is fine, but we'll use base 10 for clarity: \[ \log(15^{-6p-6}) = \log(68.2) \] Using the power rule of logarithms: \[ (-6p - 6) \log(15) = \log(68.2) \] Now, we can solve for \( p \): \[ -6p - 6 = \frac{\log(68.2)}{\log(15)} \] Now, distributing the negative sign gives us: \[ 6p + 6 = -\frac{\log(68.2)}{\log(15)} \] Dividing every term by 6: \[ p + 1 = -\frac{\log(68.2)}{6\log(15)} \] Subtracting 1: \[ p = -\frac{\log(68.2)}{6\log(15)} - 1 \] Thus, the final simplified expression for \( p \) is: \[ p = -\frac{\log(68.2)}{6\log(15)} - 1 \] Feel free to compute the values for \( \log(68.2) \) and \( \log(15) \) to find a numerical solution for \( p \).

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