Exponential and Logarithmic Functions — Test

Question 20 • 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 A coordinate plane with a horizontal x-axis and a vertical y-axis, both axes labeled from negative 5 to 5 in increments of 1 unit. Four curves are plotted, with two entirely above the x-axis and two entirely below the x-axis. Above the x-axis, one curve extends from an arrow on the y-axis at 0 comma 0.5 up to an upward-pointing arrow near 1 comma 5, and a second curve extends from an upward-pointing arrow near 1 comma 5 down to an arrow near 5 comma 0.5. Below the x-axis, one curve extends from an arrow on the y-axis at 0 comma negative 0.5 down to a downward-pointing arrow near 1 comma negative 5, and a second curve extends from a downward-pointing arrow near 1 comma negative 5 up to an arrow near 5 comma negative 0.5.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis, both axes labeled from negative 5 to 5 in increments of 1 unit. A straight line extends from a downward-pointing arrow near negative 5 comma negative 5, passes through 0 comma 0, and continues upward to an arrow near 5 comma 5. A curve extends from a downward-pointing arrow near 0 comma negative 5, increases upward to pass through 1 comma 0, and continues increasing to an arrow near 5 comma 1.5.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis, both axes labeled from negative 5 to 5 in increments of 1 unit. Two curves intersect at 1 comma 0. The first curve extends from a downward-pointing arrow near 0 comma negative 5, increases upward to pass through 1 comma 0 and 3 comma 1, and continues increasing to an arrow near 5 comma 1.5. The second curve extends from an upward-pointing arrow near 0 comma 5, decreases downward to pass through 1 comma 0 and 2 comma negative 1, and continues decreasing to an arrow near 5 comma negative 2.5

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

Both logarithms are defined only for x > 0, and both equal 0 when x = 1 because any positive base raised to the 0 power is 1. The log base 3 graph increases; for example, log₃(3) = 1. The log base 0.5 graph decreases; for example, log₀.₅(2) = −1. Choice C shows these curves intersecting at (1, 0), so it shows the solution.

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