Solve $\log _5(2 x)-\log _{\frac{1}{5}}(x)=\log _{25}(x+4)$ graphically.

To solve the equation graphically, we rearranged it into 4 x 4 − x − 4 = 0 and defined the function f ( x ) = 4 x 4 − x − 4 . Upon evaluating it at given points, the closest value to zero was found at x = 1.061 . Therefore, the solution to the logarithmic equation is approximately 1.061 .
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Answered by Anonymous | 2025-07-04

Rewrite the given logarithmic equation using logarithm properties to obtain 4 x 4 − x − 4 = 0 .
Define the function f ( x ) = 4 x 4 − x − 4 .
Evaluate f ( x ) for each of the given options: 0.504, 1.438, 0.724, and 1.061.
Find the value of x for which ∣ f ( x ) ∣ is the smallest. The solution is 1.061 ​ .

Explanation

Problem Analysis We are asked to solve the equation lo g 5 ​ ( 2 x ) − lo g 5 1 ​ ​ ( x ) = lo g 25 ​ ( x + 4 ) graphically. This means we need to find the value of x that satisfies this equation. We are given four possible solutions: 0.504, 1.438, 0.724, and 1.061. To solve this graphically, we can transform the equation into a form f ( x ) = 0 and then check which of the given values makes f ( x ) closest to zero.

Transforming the Equation First, let's rewrite the equation using logarithm properties: lo g 5 ​ ( 2 x ) − lo g 5 − 1 ​ ( x ) = lo g 5 2 ​ ( x + 4 ) lo g 5 ​ ( 2 x ) − ( − lo g 5 ​ ( x )) = 2 1 ​ lo g 5 ​ ( x + 4 ) lo g 5 ​ ( 2 x ) + lo g 5 ​ ( x ) = 2 1 ​ lo g 5 ​ ( x + 4 ) lo g 5 ​ ( 2 x 2 ) = lo g 5 ​ (( x + 4 ) 2 1 ​ ) Since the logarithms are equal, we can equate the arguments: 2 x 2 = ( x + 4 ) 2 1 ​ Squaring both sides: ( 2 x 2 ) 2 = x + 4 4 x 4 = x + 4 4 x 4 − x − 4 = 0 Let f ( x ) = 4 x 4 − x − 4 .

Evaluating the Function Now, we evaluate f ( x ) at the given possible solutions: f ( 0.504 ) = 4 ( 0.504 ) 4 − 0.504 − 4 ≈ − 4.2459 f ( 1.438 ) = 4 ( 1.438 ) 4 − 1.438 − 4 ≈ 11.6659 f ( 0.724 ) = 4 ( 0.724 ) 4 − 0.724 − 4 ≈ − 3.6250 f ( 1.061 ) = 4 ( 1.061 ) 4 − 1.061 − 4 ≈ 0.0080 We are looking for the value of x that makes f ( x ) closest to 0.

Finding the Closest Solution Comparing the absolute values of the results, we have: ∣ f ( 0.504 ) ∣ = ∣ − 4.2459∣ = 4.2459 ∣ f ( 1.438 ) ∣ = ∣11.6659∣ = 11.6659 ∣ f ( 0.724 ) ∣ = ∣ − 3.6250∣ = 3.6250 ∣ f ( 1.061 ) ∣ = ∣0.0080∣ = 0.0080 The smallest absolute value is ∣ f ( 1.061 ) ∣ = 0.0080 , which is closest to 0.

Final Answer Therefore, the graphical solution to the equation is approximately x = 1.061 .


Examples
In engineering, when designing structures or systems, it's often necessary to find the points where different factors balance each other out. For example, in electrical engineering, you might need to find the current value in a circuit where the power consumption is minimized. This involves solving equations, and graphical methods, like the one used here, can help visualize and approximate the solutions. By plotting the equation and finding where it crosses the x-axis, engineers can quickly estimate the optimal values for their designs, ensuring efficiency and stability.

Answered by GinnyAnswer | 2025-07-04