MHB Rules for Finding the Base of a Exponential Function?

AI Thread Summary
The discussion focuses on identifying the base of exponential functions, with examples provided for clarification. The base of the function f(x)=7^x is correctly identified as 7, while f(x)=3^{2x} has a base of 9 when expressed as (3^2)^x. The function f(x)=8^{\frac{4}{3}x} is analyzed, revealing that it can be rewritten as (8^{\frac{4}{3}})^x, leading to a base of 16 after simplification. However, there is a debate about whether the original base is 8 or 16, as both are discussed in the context of the transformation. Understanding the laws of exponents is crucial for determining the correct bases in these functions.
RidiculousName
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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 $$f(x)=7^x$$ is 7 and the base of $$f(x)=3^{2x}$$ is 9 but even though I know $$f(x)=8^{\frac{4}{3}x}$$ has a base of 16, I don't know how that answer was reached.
 
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You could write:

$$f(x)=8^{\Large\frac{4}{3}x}=\left(8^{\Large\frac{4}{3}}\right)^x=\left(\left(8^{\Large\frac{1}{3}}\right)^4\right)^x=\left(2^4\right)^x=16^x$$
 
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 $$f(x)=7^x$$ is 7 and the base of $$f(x)=3^{2x}$$ is 9 but even though I know $$f(x)=8^{\frac{4}{3}x}$$ has a base of 16, I don't know how that answer was reached.
Actually, the base of [math]3^{2x}[/math] is 3! Of course that is the equal to [math](3^2)^x= 9^x[/math] which base 9. [math]f(x)= 8^{\frac{4}{3}ax}[/math] has base 8. Using the "laws of exponents", [math]8^{4/3}= (8^{1/3})^4[/math] and, since [math]2^3= 8[/math], [math]8^{1/3}= 2[/math] so [math]8^{4/3}= 2^4= 16[/math] so that [math]8^{\frac{4}{3}x}= 16^x[/math].

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