Solve the equation algebraically and check graphically.
$5 e^{5 x}=1115$
$x \approx$ $\square$
(Round to the nearest thousandth as needed.)
Answer (1)
Divide both sides of the equation by 5: e 5 x = 223 .
Take the natural logarithm of both sides: 5 x = ln ( 223 ) .
Divide by 5 to isolate x : x = 5 l n ( 223 ) .
Calculate the value of x and round to the nearest thousandth: x ≈ 1.081 .
Explanation
Problem Analysis We are given the equation 5 e 5 x = 1115 and asked to solve for x , rounding to the nearest thousandth. We will solve this algebraically and then discuss how to check the solution graphically.
Isolating the Exponential First, we divide both sides of the equation by 5 to isolate the exponential term: 5 5 e 5 x = 5 1115 e 5 x = 223
Taking the Natural Logarithm Next, we take the natural logarithm of both sides of the equation to eliminate the exponential: ln ( e 5 x ) = ln ( 223 )
Simplifying Using the property of logarithms that ln ( e u ) = u , we simplify the left side of the equation: 5 x = ln ( 223 )
Solving for x Now, we divide both sides by 5 to solve for x : x = 5 ln ( 223 )
Calculating the Value of x Using a calculator, we find the value of x : x ≈ 5 5.40717 ≈ 1.081434
Rounding to Nearest Thousandth Rounding to the nearest thousandth, we get: x ≈ 1.081
Graphical Check To check graphically, we would plot the function y = 5 e 5 x and the horizontal line y = 1115 . The x-coordinate of the intersection point of these two graphs should be approximately 1.081.
Final Answer Therefore, the solution to the equation 5 e 5 x = 1115 , rounded to the nearest thousandth, is x ≈ 1.081 .
Examples
Exponential equations like this are used in various fields, such as calculating 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 such equations helps you determine how long it will take for your investment to reach a certain value. Understanding exponential growth and decay is crucial in finance, biology, and physics.
