To find the equation of the exponential curve $y = ab^x$ that passes through the points $(0, 3)$ and $(5, 70)$, we can solve for $a$ and $b$ by substituting these points into the equation.

Step 1: Use the point $(0, 3)$

Substitute $x = 0$ and $y = 3$ into the equation $y = ab^x$:

$3 = ab^0$ Since $b^0 = 1$, the equation simplifies to:

$3 = a \times 1$

Thus, we find $a = 3$.

Step 2: Use the point $(5, 70)$

Now substitute $x = 5$ and $y = 70$ into the equation $y = ab^x$. Since we already know $a = 3$, substitute this value:

$70 = 3b^5$

Solve for $b^5$:

$b^5 = \frac{70}{3} \approx 23.33$

Now solve for $b$:

$b = \left( 23.33 \right)^{1/5} \approx 2.26$

Step 3: Write the final equation

Now that we have $a = 3$ and $b \approx 2.26$, the equation of the exponential curve is:

$y = 3 \times (2.26)^x$

Final Answer:

$y = 3 \times (2.26)^x$

Verification:

To verify the result, you can plot this equation and check that it passes through the points $(0, 3)$ and $(5, 70)$. You can also check using a graphing calculator by entering the equation and seeing if the points are on the curve.

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Related questions:

1. How do you calculate the value of $b$ using logarithms?
2. How can you estimate an exponential equation when you only have two points?
3. What happens if the given points do not fit an exponential curve exactly?
4. Can you apply this method to solve for exponential curves with more than two points?
5. What other methods exist for fitting an exponential curve to data points?

Tip: Exponential growth or decay can be modeled with $y = ab^x$, where $a$ is the starting value (when $x = 0$) and $b$ is the growth factor (greater than 1 for growth, less than 1 for decay).