The discussion focuses on identifying the base of exponential functions, specifically examining functions such as $$f(x)=7^x$$, $$f(x)=3^{2x}$$, and $$f(x)=8^{\frac{4}{3}x}$$. The base of $$f(x)=7^x$$ is definitively 7, while $$f(x)=3^{2x}$$ has a base of 9, derived from the expression $$3^{2x} = (3^2)^x$$. The function $$f(x)=8^{\frac{4}{3}x}$$ simplifies to $$16^x$$, indicating a base of 16, achieved through the transformation using the laws of exponents.
PREREQUISITESStudents, educators, and anyone interested in mastering exponential functions and their properties, particularly in algebra and precalculus contexts.
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].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.