Use logarithms to solve the exponential equation. [tex]2^{x+4}=3^{x-6}[/tex] The solution is [tex]x =[/tex] $\square$ . (Simplify your answer. Round the final answer to three decimal places as needed. Round all intermediate values to six decimal places as needed.)

To solve the equation 2 x + 4 = 3 x − 6 , we take the natural logarithm of both sides and apply the power rule. This leads us to isolate x , resulting in x ≈ 23.095 when rounded to three decimal places. Thus, the solution is x = 23.095 .
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Answered by Anonymous | 2025-07-03

Take the natural logarithm of both sides of the equation: ln ( 2 x + 4 ) = ln ( 3 x − 6 ) .
Apply the power rule of logarithms: ( x + 4 ) ln ( 2 ) = ( x − 6 ) ln ( 3 ) .
Expand and rearrange the equation to isolate x : x ( ln ( 3 ) − ln ( 2 )) = 4 ln ( 2 ) + 6 ln ( 3 ) .
Solve for x and round to three decimal places: x = l n ( 3 ) − l n ( 2 ) 4 l n ( 2 ) + 6 l n ( 3 ) ​ ≈ 23.095 .

Explanation

Taking Logarithms We are given the exponential equation 2 x + 4 = 3 x − 6 . Our goal is to solve for x using logarithms. Let's take the natural logarithm (ln) of both sides of the equation.

Applying Natural Logarithm Taking the natural logarithm of both sides, we get: ln ( 2 x + 4 ) = ln ( 3 x − 6 )

Using Power Rule Using the power rule of logarithms, which states that ln ( a b ) = b ln ( a ) , we can rewrite the equation as: ( x + 4 ) ln ( 2 ) = ( x − 6 ) ln ( 3 )

Expanding the Equation Expanding both sides of the equation, we have: x ln ( 2 ) + 4 ln ( 2 ) = x ln ( 3 ) − 6 ln ( 3 )

Isolating x Now, we want to isolate x . Rearrange the terms to get all x terms on one side and the constants on the other side: x ln ( 3 ) − x ln ( 2 ) = 4 ln ( 2 ) + 6 ln ( 3 ) x ( ln ( 3 ) − ln ( 2 )) = 4 ln ( 2 ) + 6 ln ( 3 )

Solving for x Now, we solve for x by dividing both sides by ( ln ( 3 ) − ln ( 2 )) :
x = ln ( 3 ) − ln ( 2 ) 4 ln ( 2 ) + 6 ln ( 3 ) ​

Calculating the Value of x We know that ln ( 2 ) ≈ 0.693147 and ln ( 3 ) ≈ 1.098612 . Substituting these values into the equation, we get: x = 1.098612 − 0.693147 4 ( 0.693147 ) + 6 ( 1.098612 ) ​ x = 0.405465 2.772588 + 6.591672 ​ x = 0.405465 9.36426 ​ x ≈ 23.095113

Final Answer Rounding the final answer to three decimal places, we get: x ≈ 23.095


Examples
Logarithms are incredibly useful in many real-world scenarios, especially when dealing with exponential growth or decay. For instance, calculating the time it takes for an investment to double at a certain interest rate involves solving an exponential equation using logarithms. Similarly, in environmental science, logarithms are used to model the decay of radioactive substances or to determine the age of ancient artifacts through carbon dating. Understanding how to solve exponential equations with logarithms provides a powerful tool for analyzing and predicting outcomes in finance, science, and engineering.

Answered by GinnyAnswer | 2025-07-03