Exponential Functions: Practice

The base is the fixed number that gets raised to the variable power.

$f(x) = 7^x \quad\Rightarrow\quad \text{base} = 7$

The exponent is the variable $x$, and $49$ is just $7^2$, not the base.

Any allowed base raised to the power $0$ equals $1$.

$f(0) = 5^0 = 1$

This is why every graph $b^x$ passes through $(0, 1)$.

The exponent $3$ means three factors of $2$.

$2^3 = 2 \times 2 \times 2 = 8$

It is not $2 \times 3 = 6$.

Decay happens when the base lies between $0$ and $1$, so each step multiplies by less than $1$.

$0 < \tfrac{1}{3} < 1 \;\Rightarrow\; \text{decay}$

Bases $2$ and $5$ give growth, and $b = 1$ is not allowed because it is constant.

The exponent $2$ means two factors of $3$.

$3^2 = 3 \times 3 = 9$

It is not $3 \times 2 = 6$.

An exponential function keeps the variable in the exponent. In $x^2$ the variable is the base and the exponent is fixed, so it is a power function.

$x^2 \;\text{is a power function, not exponential}$

The other three all have the variable in the exponent.

The exponent $4$ means four factors of $2$.

$2^4 = 2 \times 2 \times 2 \times 2 = 16$

The base is the fixed number being raised to the variable power.

$\text{base} = \tfrac{1}{2}$

Since $0 < \tfrac{1}{2} < 1$, this base gives decay.

Doubling multiplies the amount by $2$ each hour, and a constant multiplier greater than $1$ is exponential growth.

$\times 2 \text{ each step} \;\Rightarrow\; \text{growth}$

Any allowed base raised to $0$ is $1$.

$2^0 = 1$

Two factors of $10$.

$10^2 = 10 \times 10 = 100$

A constant multiplier between steps is a constant ratio, which is the signature of an exponential function.

$\frac{6}{2} = \frac{18}{6} = \frac{54}{18} = 3$

The ratio is a steady $3$, so the pattern is exponential.