To analyze the equation log6x + 1 = log22(2x + 2), we will begin by exploring its components and how they behave when graphed.
The left-hand side, log6x + 1, represents the logarithm of x to the base 6, shifted vertically upwards by 1 unit. As x approaches 0 from the right, this function approaches negative infinity, and as x increases, the logarithmic function will rise slowly, approaching positive infinity but never quite touching it.
On the right-hand side, log22(2x + 2) represents the logarithm of (2x + 2) to the base 22. Note that this function is only defined for values of x greater than -1, since (2x + 2) must be positive. As x increases beyond -1, (2x + 2) will increase steadily, and consequently, the logarithm will increase as well. However, the rate of increase is also slower due to the base 22, which is greater than 1.
Both functions, therefore, share the characteristic of increasing as x increases, but at different rates due to their respective bases. When analyzing intersections, the point where these two graphs cross represents the solutions to the equation. The behavior of these functions indicates that:
- The graph of log6x is defined for x > 0, while log22(2x + 2) is defined for x > -1.
- At some point on the graph, it is very likely that there will exist a unique intersection since one function increases faster at lower x-values while the other gains more ground as x increases.
In summary, the true statement about the graph of the given equation is that it’s composed of two distinct logarithmic functions that each capture the behavior of their respective logarithms and yield intersections indicating the solutions of the equation.