sketch the parametric curve by eliminating the parameter.

$x={\log }_{10}t,y=\ln t,t\in (0,\infty )$

The parametric curve is$x={\log }_{10}t,y=\ln t,t\in (0,\infty )$

Let us consider the parametric equations$x={\log }_{10}{t}_{*}y=\ln t,t\in (0,\infty )$.

The objective is to sketch the parametric curve by eliminating the parameters.

Take the equation$x={\log }_{10}t$.

By using definition of logarithms we know that ,

If ${\log }_{a}x=m\Rightarrow a^n=x$

Here $x={\log }_{10}t$

${10}^x=t$since by definition of $\log$

Substitute in the equation$y=\ln t$.

Then,

$y=\ln {10}^x$

In order to draw the graph, let's assume

Substitute in the equation$y=\ln {10}^x$.

Then,

$$\begin{gathered} y=\ln {10}^0 \\ y=\ln 1 \\ y=0 \\ (x,y)=(0,0) \end{gathered}$$

Substitute x = 1 in the equation$y=\ln {10}^x$.

Then,

$$\begin{gathered} y=\ln {10}^1 \\ y=1\ln 10 \\ y=1\left(2302\right) \\ (x,y)=(1,2.302) \end{gathered}$$

Substitute $x=2$in the equation$y=\ln {10}^x$.

Then,

$$\begin{gathered} y=\ln {10}^2 \\ y=2\ln 10 \\ y=2(2.302) \\ (x,y)=(2,4.604) \end{gathered}$$

The graphical representation using the points $(0,0)(1,2.302)(2,4.604)$is as follows:

The equation after elimination of the parameter is $y=\ln {10}^x$