Solve $\log _2 x=\log _{12} x$ by graphing.
What equations should be graphed?
$y_1=\frac{\log x}{\log 2}$
$y_2=\frac{\log x}{\log 12}$
Answer (2)
Use the change of base formula to rewrite the equation: l o g 2 l o g x = l o g 12 l o g x .
Identify the equations to graph: y 1 = l o g 2 l o g x and y 2 = l o g 12 l o g x .
Graph the equations and find the intersection points.
The x-coordinate of the intersection point is the solution to the equation: x = 1 .
Explanation
Problem Analysis We are given the equation lo g 2 x = lo g 12 x and asked to solve it by graphing. We need to determine which equations should be graphed to find the solution.
Change of Base Formula Using the change of base formula, we can rewrite the logarithms in terms of a common base, such as base 10. The change of base formula is lo g a b = l o g c a l o g c b . Applying this to our equation, we get l o g 2 l o g x = l o g 12 l o g x .
Identifying the Equations Now, let's consider the given options for the equations to be graphed. We want to graph two equations, y 1 and y 2 , such that their intersection points give us the solution to the original equation. From the change of base formula, we have l o g 2 l o g x = l o g 12 l o g x . Therefore, we can set y 1 = l o g 2 l o g x and y 2 = l o g 12 l o g x .
Graphical Solution The graphs of y 1 = l o g 2 l o g x and y 2 = l o g 12 l o g x will intersect where l o g 2 l o g x = l o g 12 l o g x . This is equivalent to our original equation lo g 2 x = lo g 12 x .
Algebraic Solution To solve the equation algebraically, we can rearrange the equation l o g 2 l o g x = l o g 12 l o g x as follows: lo g 2 lo g x − lo g 12 lo g x = 0 lo g x ( lo g 2 1 − lo g 12 1 ) = 0 Since l o g 2 1 − l o g 12 1 = 0 , we must have lo g x = 0 . This implies that x = 1 0 0 = 1 . Therefore, the solution to the equation is x = 1 .
Final Answer The equations to be graphed are y 1 = l o g 2 l o g x and y 2 = l o g 12 l o g x .
Examples
Graphing logarithmic functions is useful in various real-world scenarios, such as analyzing the growth of populations or the decay of radioactive substances. For instance, if we want to compare the growth rate of two different populations, we can model their growth using logarithmic functions and graph them to visually determine which population is growing faster. Similarly, in finance, we can use logarithmic graphs to analyze the growth of investments over time and compare different investment options.
The solution to lo g 2 x = lo g 12 x can be found by graphing y 1 = l o g 10 2 l o g 10 x and y 2 = l o g 10 12 l o g 10 x . The intersection point of these two graphs is at x = 1 , which is the solution to the equation. Hence, the solution is x = 1 .
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