Use common or natural logarithms to solve the exponential equation.
[tex]3 e^{5 x+2}=12[/tex]
x=
(Round to four decimal places as needed.)
Answer (2)
To solve the equation 3 e 5 x + 2 = 12 , we first isolate the exponential term and take the natural logarithm. After simplifying and isolating x , we find that x ≈ − 0.1227 .
;
Divide both sides of the equation by 3: e 5 x + 2 = 4 .
Take the natural logarithm of both sides: 5 x + 2 = ln ( 4 ) .
Subtract 2 from both sides: 5 x = ln ( 4 ) − 2 .
Divide by 5 to solve for x : x = 5 l n ( 4 ) − 2 ≈ − 0.1227 .
− 0.1227
Explanation
Problem Setup We are given the equation 3 e 5 x + 2 = 12 and asked to solve for x , rounding to four decimal places.
Isolating the Exponential Term First, we divide both sides of the equation by 3 to isolate the exponential term: 3 3 e 5 x + 2 = 3 12 e 5 x + 2 = 4
Taking the Natural Logarithm Next, we take the natural logarithm of both sides of the equation: ln ( e 5 x + 2 ) = ln ( 4 )
Simplifying the Logarithm Using the property of logarithms that ln ( e u ) = u , we simplify the left side: 5 x + 2 = ln ( 4 )
Isolating the x Term Now, we subtract 2 from both sides: 5 x = ln ( 4 ) − 2
Solving for x Finally, we divide both sides by 5 to solve for x : x = 5 ln ( 4 ) − 2
Calculating the Value of x We calculate the value of x : x = 5 ln ( 4 ) − 2 ≈ 5 1.386294 − 2 ≈ 5 − 0.613706 ≈ − 0.122741
Rounding the Result Rounding to four decimal places, we get x ≈ − 0.1227 .
Examples
Exponential equations are used in various fields such as finance, physics, and engineering. For example, they can model population growth, radioactive decay, and compound interest. Understanding how to solve exponential equations allows us to predict future values, determine decay rates, and calculate investment returns.
