Solve $\log _3(x+1)=\log _6(5-x)$ by graphing. What equations should be graphed?
A. $y_1=\frac{\log (x+1)}{\log 3}$
B. $y_1=\frac{\log 3}{\log (x+1)}$
C. $y_2=\frac{\log 6}{\log (5-x)}$
D. $y_2=\frac{\log (5-x)}{\log 6}$
Answer (1)
Rewrite both sides of the equation using the change of base formula with base 10: lo g 3 ( x + 1 ) = l o g ( 3 ) l o g ( x + 1 ) and lo g 6 ( 5 − x ) = l o g ( 6 ) l o g ( 5 − x ) .
Define the two functions: y 1 = l o g ( 3 ) l o g ( x + 1 ) and y 2 = l o g ( 6 ) l o g ( 5 − x ) .
Graph the two functions y 1 and y 2 .
The x-coordinate of the intersection point of the two graphs is the solution to the equation: y 1 = lo g 3 lo g ( x + 1 ) , y 2 = lo g 6 lo g ( 5 − x )
Explanation
Understanding the Problem We are given the equation lo g 3 ( x + 1 ) = lo g 6 ( 5 − x ) and asked to solve it by graphing. This means we need to identify two functions, y 1 and y 2 , such that the solution to the equation is the x-coordinate of their intersection point.
Change of Base Formula To solve the equation by graphing, we need to express each side of the equation as a function of x . We can use the change of base formula to 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 .
Rewriting the Left Side Let's rewrite the left side of the equation, lo g 3 ( x + 1 ) , using the change of base formula with base 10: lo g 3 ( x + 1 ) = lo g ( 3 ) lo g ( x + 1 ) So, we can define y 1 = l o g ( 3 ) l o g ( x + 1 ) .
Rewriting the Right Side Now, let's rewrite the right side of the equation, lo g 6 ( 5 − x ) , using the change of base formula with base 10: lo g 6 ( 5 − x ) = lo g ( 6 ) lo g ( 5 − x ) So, we can define y 2 = l o g ( 6 ) l o g ( 5 − x ) .
Identifying the Correct Equations The solution to the original equation lo g 3 ( x + 1 ) = lo g 6 ( 5 − x ) is the x-coordinate of the intersection point of the two graphs y 1 = l o g ( 3 ) l o g ( x + 1 ) and y 2 = l o g ( 6 ) l o g ( 5 − x ) . Comparing these equations with the given options, we see that the correct equations to graph are y 1 = l o g 3 l o g ( x + 1 ) and y 2 = l o g 6 l o g ( 5 − x ) .
Final Answer Therefore, the equations that should be graphed are: y 1 = lo g 3 lo g ( x + 1 ) y 2 = lo g 6 lo g ( 5 − x )
Examples
Logarithmic equations are useful in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, or modeling population growth in biology. By understanding how to solve logarithmic equations graphically, we can analyze and interpret data in these fields more effectively. For example, if we know the magnitude of an earthquake and the distance from the epicenter, we can use logarithmic equations to estimate the energy released by the earthquake. Similarly, in finance, logarithmic scales are used to represent stock market indices and other financial data, allowing for easier comparison of percentage changes over time.
