We are given that $f(x)$ is an exponential function of the form $y = ab^x$. We also know that:

$f(5) = 5 \quad \text{and} \quad f(14) = 35$

The general form of the function is $f(x) = ab^x$.

Step 1: Find $a$ and $b$

We can use the given points to create two equations:

1. $5 = ab^5$
2. $35 = ab^{14}$

Step 2: Divide the equations to eliminate $a$

$\frac{f(14)}{f(5)} = \frac{ab^{14}}{ab^5} = b^{14-5} = b^9$ $\frac{35}{5} = b^9 \quad \Rightarrow \quad 7 = b^9$

Now, solve for $b$:

$b = \sqrt[9]{7} \approx 1.233$

Step 3: Substitute $b$ back into one of the original equations to find $a$

Using $5 = ab^5$:

$5 = a \times \left(1.233\right)^5$

First, calculate $1.233^5$:

$1.233^5 \approx 2.930$

Then, solve for $a$:

$5 = a \times 2.930 \quad \Rightarrow \quad a = \frac{5}{2.930} \approx 1.707$

Step 4: Find $f(18)$

Now that we have $a$ and $b$, the function is:

$f(x) = 1.707 \times 1.233^x$

Substituting $x = 18$ into the equation:

$f(18) = 1.707 \times 1.233^{18}$

Calculate $1.233^{18}$:

$1.233^{18} \approx 9.896$

Then:

$f(18) \approx 1.707 \times 9.896 \approx 16.89$

Final Answer:

The value of $f(18)$ to the nearest hundredth is approximately 16.89.

Would you like more details or have any other questions?

Here are 5 related questions:

1. How would you solve for $b$ if the function were $y = ab^{2x}$?
2. Can you find $f(x)$ for a different $x$ value using this function?
3. What happens if $f(14)$ was a different value?
4. How would the function change if $b$ was negative?
5. What is the significance of the base $b$ in exponential functions?

Tip: Exponential functions grow rapidly; small changes in the exponent result in large changes in the output.