Question

If the logarithm log₃(x) = log₀.₅(x), what does the graph of this equation look like?

a) Parallel lines
b) Perpendicular lines
c) No intersection
d) Same line

Answer 1

The equation log₃(x) = log₀.₅(x) suggests an impossibility as the two logarithmic functions have different bases, with one having a positive slope and the other a negative slope, so their graphs do not intersect.

Step-by-step explanation:

If the equation log₃(x) = log₀.₅(x) is given, we are looking at two logarithmic functions with different bases that supposedly intersect at the same value of x. To graph this equation, we need to understand the properties of logarithms and the algebra of straight lines.

However, this equation presents an impossibility. The left side of the equation represents a logarithm to the base 3, which implies a positive slope since the base is greater than 1. On the other hand, the right side represents a logarithm to the base 0.5, which implies a negative slope because the base is less than 1. According to the property of logarithms, such functions cannot intersect because they have different rates of growth, so their graphs would never cross. Therefore, the correct answer is that there would be no intersection, making the graph of this equation represent two lines that do not intersect.