Questions: Solve the exponential equation. Round to three decimal places when needed. 5(0.8^x)=4 x=

Solution Steps

To solve the exponential equation \(5(0.8^x) = 4\), we first isolate the exponential term by dividing both sides by 5, resulting in \(0.8^x = \frac{4}{5}\). Next, we take the logarithm of both sides to solve for \(x\). Using the property of logarithms that \(\log(a^b) = b \log(a)\), we can solve for \(x\) by dividing the logarithm of the right side by the logarithm of 0.8.

Starting with the equation: \[ 5(0.8^x) = 4 \] we divide both sides by 5: \[ 0.8^x = \frac{4}{5} \]

Next, we take the logarithm of both sides: \[ \log(0.8^x) = \log\left(\frac{4}{5}\right) \] Using the property of logarithms, we can rewrite the left side: \[ x \log(0.8) = \log\left(\frac{4}{5}\right) \]

Now, we solve for \(x\) by dividing both sides by \(\log(0.8)\): \[ x = \frac{\log\left(\frac{4}{5}\right)}{\log(0.8)} \] Calculating this gives: \[ x \approx 1.0 \]

Final Answer