To solve the equation [tex]\(\log_3(x+1) = \log_6(5-x)\)[/tex] by graphing, we need to identify and graph the appropriate functions to find where they intersect. The intersection points of these functions will give us the solution to the equation.
Let's break it down step-by-step:
1. Understand the equation: We need to solve [tex]\(\log_3(x+1) = \log_6(5-x)\)[/tex].
2. Express the logarithms using the change of base formula: The change of base formula for logarithms is [tex]\(\log_b(a) = \frac{\log(a)}{\log(b)}\)[/tex], where [tex]\(\log\)[/tex] represents the logarithm in base 10 or natural logarithm (base [tex]\(e\)[/tex]).
Applying this formula, we have:
[tex]\[
\log_3(x+1) = \frac{\log(x+1)}{\log(3)}
\][/tex]
And:
[tex]\[
\log_6(5-x) = \frac{\log(5-x)}{\log(6)}
\][/tex]
3. Define the functions to graph: We can rewrite our original equation in terms of two functions:
[tex]\[
y_1 = \frac{\log(x+1)}{\log(3)}
\][/tex]
[tex]\[
y_2 = \frac{\log(5-x)}{\log(6)}
\][/tex]
These are the functions we need to graph to find their intersection points.
4. Graph the functions: Plot the equations [tex]\(y_1\)[/tex] and [tex]\(y_2\)[/tex] on the same set of axes. The x-coordinate of the point(s) where these two curves intersect will be the solution to our original equation [tex]\(\log_3(x+1) = \log_6(5-x)\)[/tex].
To summarize, the correct equations to graph are:
[tex]\[ y_1 = \frac{\log(x+1)}{\log(3)} \][/tex]
and
[tex]\[ y_2 = \frac{\log(5-x)}{\log(6)} \][/tex]
By graphing these equations, you can determine the values of [tex]\(x\)[/tex] where [tex]\(y_1\)[/tex] and [tex]\(y_2\)[/tex] intersect. These [tex]\(x\)[/tex] values are the solutions to the original equation.
