Evaluating exponential functions

An exponential function looks like $f(x) = a \times b^x$, where $a$ is the starting value and $b$ is the base. To evaluate it at a given $x$, substitute that value and compute the power first, then multiply by the starting value.

For $f(x) = 5 \times 3^x$, to find $f(2)$ substitute $x = 2$: $5 \times 3^2 = 5 \times 9 = 45$.

The base is the growth factor: every time $x$ increases by $1$, the output is multiplied by $b$. For $g(x) = 3 \times 4^x$, going from $x$ to $x + 1$ multiplies the value by $4$, because $g(x+1) = 3 \times 4^{x+1} = 4 \times g(x)$.

The exponent is often a fraction of time. If a population is $p \times 2^{\frac{t}{3}}$, then at $t = 12$ the exponent is $\frac{12}{3} = 4$, so the value is $p \times 2^4 = 16p$. Choosing a time that makes the exponent a whole number keeps the arithmetic simple.

If $f(x) = 2 \times 5^x$, what is $f(3)$?
Substitute $x = 3$: $2 \times 5^3 = 2 \times 125 = 250$.

For $g(x) = 2 \times 5^x$, by what factor does the output grow as $x$ increases by $1$?
$g(x+1) = 2 \times 5^{x+1} = 5 \times (2 \times 5^x) = 5 \times g(x)$.
So the output grows by a factor of $5$.

A population is modeled by $P = 100 \times 3^{\frac{t}{2}}$, where $t$ is in years. What is the population at $t = 6$?
The exponent is $\frac{6}{2} = 3$, so $P = 100 \times 3^3 = 100 \times 27 = 2700$.