Questions: Find the exact solution to the exponential equation (41=12(0.7)^q). (Do not give a decimal approximation.) (q=) help (logarithms)

Solution Steps

To solve the exponential equation \(41 = 12(0.7)^q\), we need to isolate the variable \(q\). First, divide both sides by 12 to simplify the equation. Then, take the logarithm of both sides to solve for \(q\). Use the properties of logarithms to bring down the exponent and solve for \(q\).

Starting with the equation: \[ 41 = 12(0.7)^q \] we divide both sides by 12 to isolate the exponential term: \[ \frac{41}{12} = (0.7)^q \] Calculating the left side gives: \[ \frac{41}{12} \approx 3.4167 \]

Next, we take the logarithm of both sides: \[ \log\left(\frac{41}{12}\right) = \log\left((0.7)^q\right) \] Using the property of logarithms that allows us to bring down the exponent, we have: \[ \log\left(\frac{41}{12}\right) = q \cdot \log(0.7) \]

Now, we can solve for \(q\) by dividing both sides by \(\log(0.7)\): \[ q = \frac{\log\left(\frac{41}{12}\right)}{\log(0.7)} \] Calculating this gives: \[ q \approx -3.4448 \]

Final Answer