Final answer:
To solve the given exponential equation e²ˣ⁺⁹=10(x/11), take the natural logarithm of both sides to eliminate the exponent. Isolate x by moving terms involving x to one side and constant terms to the other side. Divide both sides by (2 - ln(10)/11) to get the exact expression for x and calculate the decimal approximation using ln(10) = 2.30259.
Step-by-step explanation:
To solve the given exponential equation e²ˣ⁺⁹=10(x/11), we need to eliminate the exponent on both sides of the equation. To do this, we can take the natural logarithm (ln) of both sides. The natural logarithm of e²ˣ⁺⁹ is simply 2x+9. The natural logarithm of 10(x/11) can be expressed as (x/11)ln(10). So our equation becomes 2x+9 = (x/11)ln(10).
Next, we can isolate x by moving the terms involving x to one side of the equation and the constant terms to the other side. Subtracting (x/11)ln(10) from both sides, we get 2x - (x/11)ln(10) = -9. Simplifying further, we can factor out x from the left side to get x(2 - ln(10)/11) = -9.
To solve for x, we divide both sides of the equation by (2 - ln(10)/11). The exact expression for x is x = -9/(2 - ln(10)/11). To find the decimal approximation, we can substitute the value of ln(10) as approximately 2.30259 and calculate the value of x using a scientific calculator or spreadsheet software.