Consider the equation [tex]9 \cdot e^{2 z}=54[/tex]. Solve the equation for [tex]z[/tex]. Express the solution as a logarithm in base-e. [tex]z=[/tex] Approximate the value of [tex]z[/tex]. Round your answer to the nearest thousandth. [tex]z \approx[/tex]

Question

GinnyAnswer · Answer

Divide both sides of the equation by 9: e 2 z = 6 . 
 Take the natural logarithm of both sides: 2 z = ln ( 6 ) . 
 Solve for z : z = 2 l n ( 6 ) ​ . 
 Approximate the value of z to the nearest thousandth: z ≈ 0.896 . 
 
 Explanation 
 
 Problem Analysis We are given the equation 9 × e 2 z = 54 and asked to solve for z , expressing the solution as a logarithm in base e and approximating the value to the nearest thousandth. 
 
 Isolating the Exponential Term First, divide both sides of the equation by 9 to isolate the exponential term: 9 9 × e 2 z ​ = 9 54 ​ e 2 z = 6 
 
 Taking the Natural Logarithm Next, take the natural logarithm (base e ) of both sides of the equation: ln ( e 2 z ) = ln ( 6 ) 
 
 Simplifying the Logarithm Using the property of logarithms, we can simplify the left side of the equation: 2 z = ln ( 6 ) 
 
 Solving for z Now, divide both sides by 2 to solve for z :
 z = 2 ln ( 6 ) ​ 
 
 Approximating the Value of z To approximate the value of z , we can use a calculator to find the value of ln ( 6 ) and then divide by 2. The result of this operation is approximately 0.8958797346140275. Rounding to the nearest thousandth, we get: z ≈ 0.896

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. Solving for variables in such equations helps in predicting future values or understanding past trends. In finance, understanding exponential growth is crucial for investment planning and assessing the impact of compound interest on savings or loans. Similarly, in physics, it helps in determining the half-life of radioactive substances.

Anonymous · Answer

To solve the equation 9 ⋅ e 2 z = 54 , we isolate the exponential term to find z as 2 l n ( 6 ) ​ . The approximate value of z rounded to the nearest thousandth is 0.896 . 
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Consider the equation [tex]9 \cdot e^{2 z}=54[/tex]. Solve the equation for [tex]z[/tex]. Express the solution as a logarithm in base-e. [tex]z=[/tex] Approximate the value of [tex]z[/tex]. Round your answer to the nearest thousandth. [tex]z \approx[/tex]

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