Find the solution of the exponential equation 15e^x-3=7 The exact solution, in terms of the natural logarithm is: x=square The approximate solution rounded to 4 decimal places is: x=square

To solve the exponential equation \(15e^x - 3 = 7\), follow these steps:<br /><br />1. **Isolate the exponential term:**<br /> \[<br /> 15e^x - 3 = 7<br /> \]<br /> Add 3 to both sides:<br /> \[<br /> 15e^x = 10<br /> \]<br /><br />2. **Solve for \(e^x\):**<br /> Divide both sides by 15:<br /> \[<br /> e^x = \frac{10}{15} = \frac{2}{3}<br /> \]<br /><br />3. **Take the natural logarithm of both sides:**<br /> \[<br /> \ln(e^x) = \ln\left(\frac{2}{3}\right)<br /> \]<br /> Using the property \(\ln(e^x) = x\):<br /> \[<br /> x = \ln\left(\frac{2}{3}\right)<br /> \]<br /><br />So, the exact solution in terms of the natural logarithm is:<br />\[<br />x = \ln\left(\frac{2}{3}\right)<br />\]<br /><br />4. **Find the approximate solution:**<br /> Use a calculator to find the numerical value of \(\ln\left(\frac{2}{3}\right)\):<br /> \[<br /> x \approx \ln\left(\frac{2}{3}\right) \approx -0.4055<br /> \]<br /><br />So, the approximate solution rounded to 4 decimal places is:<br />\[<br />x \approx -0.4055<br />\]<br /><br />Therefore, the exact solution is \(x = \ln\left(\frac{2}{3}\right)\) and the approximate solution is \(x \approx -0.4055\).