There is a question that is asking me ($N$=number of hectares destroyed) ($t$=hours since passed) where the function for finding land that is burned out is
$N=40 \log_{10}(500t+1)$
it wants me to find the amount of time ($t$) until the hectares (land) burned is $155$. How do i work backwards to get the time until the specified land amount is burned out.
I just want to know any function or how to use this function to get the time until the specified amount of hectares ($155$) is burned out. Thank you.
You have
$$N = 40\log_{10}(500t + 1) \tag{1}\label{eq1}$$
You can rearrange this to get $t$ in terms of $N$ by dividing, taking the values to the power of $10$, and a few other manipulations, to get
$$\begin{equation}\begin{aligned} \frac{N}{40} & = \log_{10}(500t + 1) \\ 10^{N/40} & = 500t + 1 \\ 10^{N/40} - 1 & = 500t \\ t & = \frac{10^{N/40} - 1}{500} \end{aligned}\end{equation}\tag{2}\label{eq2}$$
With $N = 155$, \eqref{eq2} gives $t = 14.99\ldots$ , i.e., basically $15$ hours.