Find the solution of the exponential equation \( e^{2 x+1}=29 \) in terms of logarithms, or correct to four decimal places. \( x=\square \) Question Help: Video

To solve the equation \( e^{2x+1} = 29 \), we start by taking the natural logarithm of both sides: \[ 2x + 1 = \ln(29) \] Next, we isolate \( x \) by rearranging the equation: \[ 2x = \ln(29) - 1 \] Finally, divide by 2: \[ x = \frac{\ln(29) - 1}{2} \] Calculating this gives approximately \( x \approx 0.8951 \) when rounded to four decimal places. This equation reflects how exponential growth can be modeled using logarithms, making it easier to find unknown values. In practical terms, understanding such equations is fundamental in fields ranging from finance to biology, where growth processes are modeled using exponential functions.