Use natural logarithms to solve the equation.
[tex]$6 e^{2 x}-9=23$[/tex]
Round to the nearest thousandth.
Answer (1)
Isolate the exponential term by adding 9 to both sides and dividing by 6: e 2 x = 6 32 .
Take the natural logarithm of both sides: 2 x = ln ( 6 32 ) .
Solve for x by dividing by 2: x = 2 l n ( 6 32 ) .
Approximate the value of x to the nearest thousandth: 0.837 .
Explanation
Problem Setup We are given the equation 6 e 2 x − 9 = 23 and we want to solve for x using natural logarithms. We will isolate the exponential term, take the natural logarithm of both sides, and then solve for x .
Isolating the Exponential Term First, we add 9 to both sides of the equation: 6 e 2 x − 9 + 9 = 23 + 9
6 e 2 x = 32
Further Isolation Next, we divide both sides by 6: 6 6 e 2 x = 6 32
e 2 x = 6 32 = 3 16
Applying Natural Logarithm Now, we take the natural logarithm of both sides: ln ( e 2 x ) = ln ( 3 16 )
Simplifying the Logarithm Using the property of logarithms, ln ( e 2 x ) = 2 x , so we have: 2 x = ln ( 3 16 )
Solving for x Finally, we divide by 2 to solve for x : x = 2 ln ( 3 16 )
Approximating the Solution We can approximate the value of x using a calculator: x = 2 ln ( 3 16 ) ≈ 2 1.686 ≈ 0.837
Final Answer Therefore, the solution to the equation, rounded to the nearest thousandth, is x ≈ 0.837 .
Examples
Exponential equations are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds interest continuously, the amount of money you have after a certain time can be modeled by an exponential equation. Solving these equations helps you determine how long it will take for your investment to reach a certain value.
