# Analytic expression for a transformed function

In MATLAB, I plotted an exponential distribution (p.d.f.) in time $\lambda e^{-\lambda t}$ but set the $x$ and $y$ axes to logarithmic. The plot looks like below.

I'm trying to mathematically find the equation of the curve. I thought setting the $x$ axis to log would obey the transformation $t\rightarrow log(t)$ and setting the $y$ axis to log would just be applying the log to the resulting function. But that was not it - I plotted the resulting function on normal axes to verify.

• I meant what equation when plotted on linear $x$ and $y$ axes looks like the curve in the figure above, up to a proportionality constant. – IanDsouza Oct 9 '17 at 7:10
• I see, in this case @Harry49 already gave you the answer. Just plot $$f(t) = \log\lambda - \frac{\lambda}{\ln 10}10^t$$ on a linear scale – caverac Oct 9 '17 at 11:50
One has \begin{aligned} \log \left(\lambda e^{-\lambda t}\right) &= \log\lambda + \frac{\ln e^{-\lambda t}}{\ln 10} \\ &= \log\lambda - \frac{\lambda t}{\ln 10} \\ & = \log\lambda -\frac{\lambda}{\ln 10} 10^{\log t} \end{aligned} The dependence of $\log \left(\lambda e^{-\lambda t}\right)$ with respect to $\log t$ is exponential. However, $\log \left(\lambda e^{-\lambda t}\right)$ is linear with respect to $t$.