Solve the exponential equation. [tex]5^{7 x-2}=8^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] [ ] .
(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: ln ( 5 7 x − 2 ) = ln ( 8 x ) .
Apply the power rule of logarithms: ( 7 x − 2 ) ln ( 5 ) = x ln ( 8 ) .
Isolate x: x ( 7 ln ( 5 ) − ln ( 8 )) = 2 ln ( 5 ) .
Solve for x: x = 7 l n ( 5 ) − l n ( 8 ) 2 l n ( 5 ) ≈ 0.350 .
Explanation
Problem Analysis We are given the exponential equation 5 7 x − 2 = 8 x . Our goal is to solve for x . To do this, we will take the natural logarithm of both sides of the equation.
Applying Logarithms Taking the natural logarithm (ln) of both sides, we get ln ( 5 7 x − 2 ) = ln ( 8 x ) . Using the power rule of logarithms, we can rewrite this as ( 7 x − 2 ) ln ( 5 ) = x ln ( 8 ) .
Isolating x Terms Expanding the left side, we have 7 x ln ( 5 ) − 2 ln ( 5 ) = x ln ( 8 ) . Now, we want to isolate the terms with x on one side of the equation. So, we rearrange the equation to get 7 x ln ( 5 ) − x ln ( 8 ) = 2 ln ( 5 ) .
Solving for x Factoring out x from the left side, we have x ( 7 ln ( 5 ) − ln ( 8 )) = 2 ln ( 5 ) . Now, we can solve for x by dividing both sides by ( 7 ln ( 5 ) − ln ( 8 )) , which gives us x = 7 l n ( 5 ) − l n ( 8 ) 2 l n ( 5 ) .
Approximating the Solution Now, we need to approximate the value of x . Using a calculator, we find that x ≈ 0.350 .
Examples
Exponential equations are used in various fields such as finance, biology, and physics. 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. In finance, we can use exponential equations to calculate the future value of an investment with compound interest. In biology, we can model the growth of a bacterial population. In physics, we can describe the decay of a radioactive substance.
The solution to the exponential equation 5 7 x − 2 = 8 x is approximately x ≈ 0.350 .
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