Find the exact value of x from the equation 3^x * e^4x = e^7

To begin to solve this equation we must take natural logarithms of both sides of the equation. This gives: ln(3^x*e^4x) = lne^7 Then we can use the log rules on the left hand side to expand it slightly to: ln3^x + lne^4x = lne^7 We can then bring down the powers for all these logarithms to give: xln3 + 4xlne = 7lne We know that lne = 1 as lne means e to what power gives e? The answer is therefore 1 = lne This gives us from the previous equation: xln3 + 4x = 7 Now we use simply rearrangement to give: x(4 + ln3) = 7 x = 7/(4 + ln3)