Sketch the graph of $y=\lg 2x$. In order to solve the equation ${{10}^{x}}=\frac{1000}{{{x}^{2}}}$, a suitable straight line has to be drawn on the same diagram as the graph of $y=\lg 2x$. Find the equation of the straight line and the number of solutions to the equation.

Exponential and Logarithmic Functions - Sketching Logarithmic Functions and Solving Equations Graphically

IP 3

Sketch the graph of $y=\lg 2x$.
In order to solve the equation ${{10}^{x}}=\frac{1000}{{{x}^{2}}}$, a suitable straight line has to be drawn on the same diagram as the graph of $y=\lg 2x$. Find the equation of the straight line and the number of solutions to the equation.

Solution:

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Answer:(i) Increasing $y=\lg(2x)$, $x>0$, through $\left(\frac12,0\right)$, asymptote $x=0$. (ii) Add $y=\frac32+\lg2-\frac x2$; exactly $1$ real solution.

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