Use logarithms to solve the exponential equation. [tex]2^{x+2}=5^{x-4}[/tex] The solution is [tex]x=[/tex] ____. Simplify your answer. Round the final answer to three decimal places as needed. Round all intermediate values to six decimal places as needed.

Take the natural logarithm of both sides: ln ( 2 x + 2 ) = ln ( 5 x − 4 ) .
Apply the power rule: ( x + 2 ) ln ( 2 ) = ( x − 4 ) ln ( 5 ) .
Expand and rearrange to isolate x : x ( ln ( 2 ) − ln ( 5 )) = − 4 ln ( 5 ) − 2 ln ( 2 ) .
Solve for x and round to three decimal places: x = l n ( 5 ) − l n ( 2 ) 4 l n ( 5 ) + 2 l n ( 2 ) ​ ≈ 8.540 ​ .

Explanation

Problem Setup We are given the exponential equation 2 x + 2 = 5 x − 4 and asked to solve for x using logarithms. We will take the natural logarithm of both sides of the equation and use properties of logarithms to isolate x .

Taking Logarithms Taking the natural logarithm (ln) of both sides of the equation, we get: ln ( 2 x + 2 ) = ln ( 5 x − 4 )

Applying Power Rule Using the power rule of logarithms, which states that ln ( a b ) = b ln ( a ) , we can rewrite the equation as: ( x + 2 ) ln ( 2 ) = ( x − 4 ) ln ( 5 )

Expanding the Equation Expanding both sides of the equation, we have: x ln ( 2 ) + 2 ln ( 2 ) = x ln ( 5 ) − 4 ln ( 5 )

Isolating x Terms Now, we want to isolate the terms containing x on one side and the constant terms on the other side. Subtracting x ln ( 5 ) from both sides and subtracting 2 ln ( 2 ) from both sides, we get: x ln ( 2 ) − x ln ( 5 ) = − 4 ln ( 5 ) − 2 ln ( 2 )

Factoring out x Factoring out x from the left side, we have: x ( ln ( 2 ) − ln ( 5 )) = − 4 ln ( 5 ) − 2 ln ( 2 )

Solving for x To solve for x , we divide both sides by ( ln ( 2 ) − ln ( 5 )) :
x = ln ( 2 ) − ln ( 5 ) − 4 ln ( 5 ) − 2 ln ( 2 ) ​

Simplifying the Expression We can multiply both the numerator and the denominator by -1 to simplify the expression: x = ln ( 5 ) − ln ( 2 ) 4 ln ( 5 ) + 2 ln ( 2 ) ​

Calculating the Value of x Now, we can approximate the value of x using the values of ln ( 2 ) and ln ( 5 ) . We know that ln ( 2 ) ≈ 0.693147 and ln ( 5 ) ≈ 1.609438 . Substituting these values into the equation, we get: x = 1.609438 − 0.693147 4 ( 1.609438 ) + 2 ( 0.693147 ) ​ = 0.916291 6.437752 + 1.386294 ​ = 0.916291 7.824046 ​ ≈ 8.539717

Rounding the Answer Rounding the value of x to three decimal places, we get: x ≈ 8.540


Examples
Logarithms are incredibly useful in many real-world scenarios, such as calculating the magnitude of earthquakes using the Richter scale, determining the pH levels of chemical solutions, and modeling population growth. In finance, logarithms help in calculating continuously compounded interest and analyzing investment returns. Solving exponential equations with logarithms allows us to understand and predict phenomena that grow or decay exponentially, providing valuable insights in various scientific and economic fields.

Answered by GinnyAnswer | 2025-07-03

The solution to the exponential equation 2 x + 2 = 5 x − 4 using logarithms is approximately x = 8.540 . This is derived by taking the natural logarithm of both sides, applying properties of logarithms, and isolating x .
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Answered by Anonymous | 2025-07-04