The diagram shows the curve with equation $y = ext{log}_3 x$. (a) On the diagram in your answer booklet, sketch the curve with equation $y = 1 - ext{log}_3 x$. (b) Determine the exact value of the $x$-coordinate of the point of intersection of the two curves.

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The diagram shows the curve with equation $y = ext{log}_3 x$ - Scottish Highers Maths - Question 11 - 2018

Question 11

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The diagram shows the curve with equation $y = ext{log}_3 x$. (a) On the diagram in your answer booklet, sketch the curve with equation $y = 1 - ext{log}_3 x$. (... show full transcript

Worked Solution & Example Answer:The diagram shows the curve with equation $y = ext{log}_3 x$ - Scottish Highers Maths - Question 11 - 2018

Step 1

Sketch the curve with equation $y = 1 - ext{log}_3 x$

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Answer

To sketch the curve, first recognize that the graph of $y = 1 - ext{log}_3 x$ is derived from the graph of $y = ext{log}_3 x$ .

The curve is a reflection of $y = ext{log}_3 x$ across the line $y = 1$ .
The function decreases and intercepts the y-axis at (1,0).
The x-axis is approached but never reaches as $x$ approaches infinity.

Step 2

Determine the exact value of the $x$-coordinate of the point of intersection of the two curves

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Answer

To find the point of intersection, set the two equations equal:

$ext{log}_3 x = 1 - ext{log}_3 x$

Rearranging gives:

$2 ext{log}_3 x = 1$

Dividing by 2:

$ext{log}_3 x = rac{1}{2}$

Exponentiating both sides:

$x = 3^{ rac{1}{2}} = rac{ ext{√}3}{3}$

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