Solve the exponential equation
[tex]$2^{4 x-2}=6^x$[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is [tex]$x \approx$[/tex] $\square$
(Type an integer or decimal rounded to three decimal places as needed.)
B. The solution is not a real number.
Answer (2)
Take the natural logarithm of both sides of the equation.
Apply the power rule of logarithms and rearrange the equation to isolate x .
Solve for x to get x = 4 l n ( 2 ) − l n ( 6 ) 2 l n ( 2 ) .
Approximate the value of x to three decimal places: x ≈ 1.413 .
Explanation
Problem Analysis We are given the exponential equation 2 4 x − 2 = 6 x . Our goal is to solve for x . To do this, we will take the natural logarithm of both sides.
Applying Logarithms Taking the natural logarithm of both sides, we get ln ( 2 4 x − 2 ) = ln ( 6 x ) . Using the power rule of logarithms, we have ( 4 x − 2 ) ln ( 2 ) = x ln ( 6 ) .
Isolating x Expanding the left side, we get 4 x ln ( 2 ) − 2 ln ( 2 ) = x ln ( 6 ) . Now, we rearrange the equation to isolate x : 4 x ln ( 2 ) − x ln ( 6 ) = 2 ln ( 2 ) .
Solving for x Factoring out x , we have x ( 4 ln ( 2 ) − ln ( 6 )) = 2 ln ( 2 ) . Solving for x , we get x = 4 l n ( 2 ) − l n ( 6 ) 2 l n ( 2 ) .
Simplifying the Expression We can simplify this expression: x = l n ( 2 4 ) − l n ( 6 ) 2 l n ( 2 ) = l n ( 16 ) − l n ( 6 ) 2 l n ( 2 ) = l n ( 6 16 ) 2 l n ( 2 ) = l n ( 3 8 ) 2 l n ( 2 ) .
Approximating the Value of x Now, we approximate the value of x to three decimal places. x ≈ 1.413 .
Final Answer Therefore, the solution is x ≈ 1.413 .
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 make predictions and analyze trends in these areas. Imagine you are investing money in a bank account with a fixed interest rate. The amount of money you have after a certain period can be modeled using an exponential equation. By solving the equation, you can determine how long it will take for your investment to reach a specific target amount. This knowledge helps you make informed financial decisions and plan for your future.
The solution to the equation 2 4 x − 2 = 6 x is approximately x ≈ 1.415 .
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