Find $t$ in $N = b \times g^t$.

Question

The problem is the following:

Find the value of $t$ in $N = b × g^t$.

So for example "$512.000 = 2000 × 2^t$"

I'm not really a mathematician so their may be a simple way or it could be hard.

Answer 1

$$\begin{align*} \rm N=b\cdot g^{\rm\displaystyle t}&\iff g^{\rm\displaystyle t}=\rm\dfrac Nb\\ &\iff\color{grey}{\boxed{\color{black}{\rm\, t=\log_{\rm\displaystyle g}\left(\rm \dfrac Nb\right)}}} \end{align*}$$

where $\rm\log_{\rm\displaystyle g}$ is the logarithmic function to the base $\rm g$ which is basically the function that tells us for which number $x$, $b^x$ will give you $c$.

Example: $\rm 512\,000 = 2000 \times 2^t$. By our result, and by setting $\rm b=2000$, $\rm g=2$ and $\rm N=512\,000$, we have: $$\begin{align*} \rm t &=\log_2\left(\dfrac{512\,000}{2000}\right)\\ &= \log_2(256)\\ &=8. \end{align*}$$ You can even try it: $2000\times 2^8=2000 \times 256=512\,000$.

I hope this helps.
Best wishes, $\mathcal H$akim.