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If the graphs of the equations $y = \log_3x$ and $y = 2$ are drawn on the same set of axes, they will intersect where $x$ is equal to $\_\_\_\_$ .

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Identify direct variation and inverse variation

Full solution

Q. If the graphs of the equations

y = \log_3x

and

y = 2

are drawn on the same set of axes, they will intersect where

x

is equal to

\_\_\_\_

.

Find Intersection Point: Set the two equations equal to each other to find the intersection point. $\newline$ Since both equations are equal to $y$ , we can set them equal to each other to find the $x$ -value where they intersect. $\newline$ $\log_{3}x = 2$
Convert to Exponential Form: Convert the logarithmic equation to exponential form to solve for $x$ . To convert from logarithmic form to exponential form, we use the definition of a logarithm: if $\log_{b}a = c$ , then $b^{c} = a$ . $3^{2} = x$
Calculate x Value: Calculate the value of x. $\newline$ $3^2 = 9$ $\newline$ So, $x = 9$
Verify Solution: Verify the solution by plugging the value of $x$ back into the original equations. $\newline$ For $y=\log_{3}x$ , when $x=9$ : $\newline$ $y = \log_{3}9 = \log_{3}(3^2) = 2$ $\newline$ For $y=2$ : $\newline$ $y = 2$ $\newline$ Both equations give $y = 2$ when $x = 9$ , so the solution is correct.

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