Exponential and Logarithmic Functions — Test

Question 8 • Algebra II

Which graph shows the solution to the equation? the log base 3 of x is equal to the log base 0 point 5 of x

Answer

Image with description On a coordinate plane the x axis lists from negative 5 to 5 by 1 and y axis lists from negative 5 to 5 by 1. There are two hyperbolas defined for positive values of x. One of the parabola has vertex (3 comma 0) and it approaches the x axis and the line x equals 1. The other hyperbola has vertex at (2 comma negative 1) and it approaches the x axis and the line x equals 1.

Image with description On a coordinate plane the x axis lists from negative 5 to 5 by 1 and y axis lists from negative 5 to 5 by 1. There is a smooth curve that passes through (0 point 5 comma negative 1), and (1 comma 0), and approaches y axis in the fourth quadrant. There is a line that passes through the origin, (1 comma 1), and (2 comma 2).

Image with description On a coordinate plane the x axis lists from negative 5 to 5 by 1 and y axis lists from negative 5 to 5 by 1. Two curves are drawn. One of the curve approaches the y axis in the first quadrant and passes through (2 comma negative 1) and (4 comma negative 2). The other curve passes through (3 comma 1), (1 comma 0), and approaches y axis in the fourth quadrant.

Image with description On a coordinate plane the x axis lists from negative 5 to 5 by 1 and y axis lists from negative 5 to 5 by 1. There is a curved graph that passes through (negative 4 comma negative 2), and (negative 2 comma negative 1). It increases in the third quadrant, changes direction at the origin, and then again increases in the first quadrant passing through (5 comma 1 point 5). There is line that passes through the origin, (1 comma 1), and (2 comma 2).

Explanation

The two logarithmic expressions are equal where their graphs intersect. Both have value 0 at x = 1, since log base 3 of 1 and log base 0.5 of 1 are both 0. The graph of log base 3 of x increases through (3, 1), while log base 0.5 of x decreases through (2, −1) and (4, −2); choice C shows both curves.

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