To solve the equation [tex]\(\log_2 x = \log_{12} x\)[/tex] by graphing, we need to express both sides of the equation in terms of common logarithms (logarithms with the same base).
Recall the change of base formula for logarithms:
[tex]\[ \log_b a = \frac{\log_k a}{\log_k b} \][/tex]
Here, [tex]\( b \)[/tex] and [tex]\( k \)[/tex] are any positive numbers, and typically [tex]\( k \)[/tex] is chosen to be 10 (common logarithm) or [tex]\( e \)[/tex] (natural logarithm).
### Steps to Determine the Equations to Graph
1. Express [tex]\(\log_2 x\)[/tex] using the change of base formula:
[tex]\[ \log_2 x = \frac{\log x}{\log 2} \][/tex]
2. Express [tex]\(\log_{12} x\)[/tex] using the change of base formula:
[tex]\[ \log_{12} x = \frac{\log x}{\log 12} \][/tex]
3. Rewrite the original equation [tex]\(\log_2 x = \log_{12} x\)[/tex] using these expressions:
[tex]\[ \frac{\log x}{\log 2} = \frac{\log x}{\log 12} \][/tex]
For this to hold, the numerators (both [tex]\(\log x\)[/tex]) and the denominators ([tex]\(\log 2\)[/tex] and [tex]\(\log 12\)[/tex]) must satisfy the relation.
Therefore, the relevant mathematical expressions for graphing purposes are:
[tex]\[ y_1 = \frac{\log 2}{\log x} \][/tex]
[tex]\[ y_2 = \frac{\log x}{\log 12} \][/tex]
### Equations to Graph
From the equations we've derived:
- [tex]\( y_1 = \frac{\log 2}{\log x} \)[/tex]
- [tex]\( y_2 = \frac{\log x}{\log 12} \)[/tex]
These are the equations you need to graph in order to find the intersection point, which would give you the solution to the original equation [tex]\(\log_2 x = \log_{12} x\)[/tex].
To summarize:
- Graph [tex]\( y_1 = \frac{\log 2}{\log x} \)[/tex]
- Graph [tex]\( y_2 = \frac{\log x}{\log 12} \)[/tex]
The point where these two graphs intersect is the solution to the equation [tex]\(\log_2 x = \log_{12} x\)[/tex].
