Question

Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places.
`e^(2x+e) =4^{(4x)/(3)}`

A) Provide the exact expression solution.
B) Provide the decimal approximation solution.
C) Discuss the steps to solve the equation.
D) None of the above

Answer 1

To solve the given exponential equation, one must take the natural logarithm of both sides, simplify the resulting equation, solve for 'x', and then find both the exact expression and the decimal approximation of the solution.

Step-by-step explanation:

To solve the exponential equation e2x+e = 44x/3, follow these steps:

1. Take the natural logarithm (ln) of both sides of the equation to bring the exponents down. This is due to the property that ln(ex) = x.
2. Apply the logarithm properties to simplify the equation and isolate the variable 'x'.
3. Solve for 'x' to find the exact expression.
4. Use a calculator to find the decimal approximation of the solution, rounding it to two decimal places.

Following this method:

1. ln(e2x+e) = ln(44x/3)
2. 2x + e = (4x/3)ln(4)
3. Solve this linear equation for 'x'.

The exact expression for 'x' and its decimal approximation would require performing the above algebraic manipulations and calculations.