Is it possible to solve equlaity between two exponential functions? I was watching this video:
Comparing exponential and linear function
And Sal solved the problem by using a table where he calculates the values for each year, which got me thinking about whether there is an easier way to solve it using a formula?
I tried writing on a paper:
$$10000+5000n = 500\cdot 2^n$$
But didn't know how to solve it, or if it was the right way to do so.
I'm still going through Algebra 1 so this might be advanced for me, but I'd like to know whether it can be solved using an easier way or not.
 A: We can write Company A's payment like this:
$$10000+5000t, \text{where t is time in months}$$
Similarly, we can write Company B's payment like this:
$$500*2^t$$
Now you may be wondering, how did I get these equations?
Well, according to the problem,

Company A is offering $\$10000$ for the first month and will increase the amount each month by $\$5000$.

So, the original payment is $\$10000$, a constant that doesn't depend on a variable of time. However, the payment increases by a steady $\$5000$ every month. This is dependent on time, so we can write it as a linear equation, where the slope represents the increase in payment by a steady increment of $\$5000$.
Now, here's the second part:

Company B is offering $\$500$ for the first month and will double their payment every month.

This is an example of "exponential" growth. Why "exponential"? Because the amount of money $\it multiplies$ on itself every month. For example, if the payment was $\$500$, then it will double to $\$1000$ in the next month, then $\$2000$ the next month, then $\$4000$ and so forth. This type of growth is $\it exponential$ because it grows like an exponent. The first payment was 500, so we would write that down as a constant $500$. However, as $t$ increases, the payment doubles, so we multiply by $2$ to the power of $t$. Like any exponent, $t$, in this case, represents how many times the payment has been doubled, and since it doubles every month, this equation would correctly represent Company B's payment. You can also check that the equation is correct by plugging in $t=0$. $500 * 2^0 = 500*1 = 500$, which gives you that original payment of $\$500$.
You are trying to solve for equality between the two equations, so set them equal to each other.
$$10000+ 5000t = 500 * 2^t$$
Dividing each side by $500$, we get this:
$$20+10t = 2^t$$
This equation will not solve nicely (in terms of integer answers), but the key here is that exponential functions increase much faster than linear equations.
