What is the solution of the equation [tex]2^x=7[/tex]? Round your answer to the nearest ten-thousandth.
Answer (1)
Take the logarithm of both sides of the equation 2 x = 7 .
Use the property of logarithms to rewrite the equation as x = l o g ( 2 ) l o g ( 7 ) .
Calculate the value of x as approximately 2.807354922.
Round the value of x to the nearest ten-thousandth, resulting in 2.8074 .
Explanation
Understanding the Problem We are given the equation 2 x = 7 and asked to find the value of x , rounded to the nearest ten-thousandth. To solve for x , we can take the logarithm of both sides of the equation.
Applying Logarithms Taking the logarithm base 2 of both sides, we have: lo g 2 ( 2 x ) = lo g 2 ( 7 ) Using the property of logarithms, we get: x = lo g 2 ( 7 ) Alternatively, we can use the natural logarithm (ln) or the common logarithm (log base 10). Using the natural logarithm, we have: ln ( 2 x ) = ln ( 7 ) x ⋅ ln ( 2 ) = ln ( 7 ) x = ln ( 2 ) ln ( 7 ) Similarly, using the common logarithm, we have: lo g ( 2 x ) = lo g ( 7 ) x ⋅ lo g ( 2 ) = lo g ( 7 ) x = lo g ( 2 ) lo g ( 7 )
Calculating the Value of x Now, we calculate the value of x :
x = lo g ( 2 ) lo g ( 7 ) ≈ 2.807354922 Rounding to the nearest ten-thousandth (four decimal places), we get: x ≈ 2.8074
Final Answer Therefore, the solution to the equation 2 x = 7 , rounded to the nearest ten-thousandth, is approximately 2.8074.
Examples
Exponential equations like 2 x = 7 are used in various fields such as finance, biology, and computer science. For instance, in finance, it can model the growth of an investment with compound interest. If you invest an amount that doubles every year, you can use this equation to find out how many years it takes for your investment to reach a certain value. In biology, it can model population growth. In computer science, it can be used to analyze the time complexity of algorithms.
