# Rules for Finding the Base of a Exponential Function?

• MHB
• RidiculousName
In summary, when finding the base of an exponential function, you can use the laws of exponents to simplify the expression. In the example given, f(x)=8^{\frac{4}{3}x} can be simplified to (8^{\frac{1}{3}})^4, which is equal to 16. Therefore, the base of f(x)=8^{\frac{4}{3}x} is 16.
RidiculousName
I was wondering if anyone could point me to a set of rules for finding the base of an exponential function? I can figure out that the base of $$\displaystyle f(x)=7^x$$ is 7 and the base of $$\displaystyle f(x)=3^{2x}$$ is 9 but even though I know $$\displaystyle f(x)=8^{\frac{4}{3}x}$$ has a base of 16, I don't know how that answer was reached.

You could write:

$$\displaystyle f(x)=8^{\Large\frac{4}{3}x}=\left(8^{\Large\frac{4}{3}}\right)^x=\left(\left(8^{\Large\frac{1}{3}}\right)^4\right)^x=\left(2^4\right)^x=16^x$$

RidiculousName said:
I was wondering if anyone could point me to a set of rules for finding the base of an exponential function? I can figure out that the base of $$\displaystyle f(x)=7^x$$ is 7 and the base of $$\displaystyle f(x)=3^{2x}$$ is 9 but even though I know $$\displaystyle f(x)=8^{\frac{4}{3}x}$$ has a base of 16, I don't know how that answer was reached.
Actually, the base of $$\displaystyle 3^{2x}$$ is 3! Of course that is the equal to $$\displaystyle (3^2)^x= 9^x$$ which base 9. $$\displaystyle f(x)= 8^{\frac{4}{3}ax}$$ has base 8. Using the "laws of exponents", $$\displaystyle 8^{4/3}= (8^{1/3})^4$$ and, since $$\displaystyle 2^3= 8$$, $$\displaystyle 8^{1/3}= 2$$ so $$\displaystyle 8^{4/3}= 2^4= 16$$ so that $$\displaystyle 8^{\frac{4}{3}x}= 16^x$$.

But, again, I would say that the bases have changed. The base in $$\displaystyle 8^{\frac{4}{3}x}$$ is 8 and the base in $$\displaystyle 16^x$$ is 16.

## 1. What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants and x is the independent variable. The base, b, is a positive number greater than 1, and as the value of x increases, the value of f(x) grows at an increasing rate.

## 2. How do I find the base of an exponential function?

To find the base of an exponential function, you can use the following steps:

1. Make sure the function is in the form of f(x) = ab^x.
2. Choose two points on the function, (x1, y1) and (x2, y2).
3. Substitute the values into the function to create two equations: y1 = ab^x1 and y2 = ab^x2.
4. Divide the two equations to eliminate the constant a: y2/y1 = b^(x2-x1).
5. Take the logarithm of both sides to solve for b: log(y2/y1) = (x2-x1)log(b).
6. Solve for b: b = 10^(log(y2/y1) / (x2-x1)).

## 3. Can I use any two points on the function to find the base?

Yes, as long as the points are not on the horizontal axis (x-axis). This is because the logarithm of 0 is undefined, so you cannot use points with a y-value of 0.

## 4. Are there any other methods for finding the base of an exponential function?

Yes, you can also use logarithms to find the base of an exponential function. The steps for this method are as follows:

1. Make sure the function is in the form of f(x) = ab^x.
2. Take the logarithm of both sides of the equation: log(f(x)) = log(ab^x).
3. Use the power rule of logarithms to separate the exponent: log(f(x)) = log(a) + xlog(b).
4. Solve for log(b): log(b) = (log(f(x)) - log(a)) / x.
5. Finally, solve for b: b = 10^(log(b)).

## 5. Why is it important to find the base of an exponential function?

Finding the base of an exponential function is important because it helps us understand the behavior and growth rate of the function. The base determines how quickly the function grows or decays, and it can also affect the shape of the graph. By finding the base, we can make predictions and draw conclusions about the function.

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