Consider the equation [tex]$0.3 \cdot e^{3 x}=27$[/tex].
Solve the equation for [tex]$x$[/tex]. Express the solution as a logarithm in base-e.
[tex]$x=$[/tex]
Approximate the value of [tex]$x$[/tex]. Round your answer to the nearest thousandth.
[tex]$x \approx$[/tex]
Answer (1)
Divide both sides of the equation by 0.3: e 3 x = 90 .
Take the natural logarithm of both sides: 3 x = ln ( 90 ) .
Solve for x : x = 3 l n ( 90 ) .
Approximate the value of x : x ≈ 1.500 .
x = 3 ln ( 90 ) ≈ 1.500
Explanation
Problem Analysis We are given the equation 0.3 ⋅ e 3 x = 27 and asked to solve for x . We need to express the solution as a logarithm in base e and then approximate the value of x to the nearest thousandth.
Isolating the Exponential Term First, we divide both sides of the equation by 0.3 to isolate the exponential term: 0.3 0.3 ⋅ e 3 x = 0.3 27 e 3 x = 90
Taking the Natural Logarithm Next, we take the natural logarithm (base e ) of both sides of the equation: ln ( e 3 x ) = ln ( 90 )
Simplifying the Logarithm Using the property of logarithms that ln ( e a ) = a , we simplify the left side of the equation: 3 x = ln ( 90 )
Solving for x Now, we divide both sides by 3 to solve for x :
x = 3 ln ( 90 )
Approximating the Value of x To approximate the value of x , we can use a calculator to find the value of ln ( 90 ) and then divide by 3 . We find that ln ( 90 ) ≈ 4.4998 . Therefore, x ≈ 3 4.4998 ≈ 1.49993 Rounding to the nearest thousandth, we get x ≈ 1.500 .
Final Answer Therefore, the exact solution is x = 3 l n ( 90 ) and the approximate solution is x ≈ 1.500 .
Examples
Exponential equations like the one we solved are used in various fields, such as calculating the growth of populations, modeling radioactive decay, and determining the time it takes for an investment to double at a certain interest rate. For example, if a bacterial population starts with 1000 cells and doubles every hour, the population size after t hours can be modeled by P ( t ) = 1000 ⋅ e k t , where k is a constant related to the growth rate. Solving exponential equations helps us predict and understand these real-world phenomena.
