Exponential Function Question

[0313phd]

Problem: F is an exponential function defined by f(x)= ab to the x power. In this problem, a and b are positive constants.
If, f(5) is equal to 96 and f(7) is equal to 384, what is the value of a?

I am having a hard time conceptualizing this one, but (b/a) to the x power=96 and (b/a) to the x power equal to 384. 96 is also one one-fourth of 384, and maybe one can do something with that, I don't know. The difference between 96 and 384 is 288, and I don't know what to do with that either. Are these two equations with more than one of the same variables? But I am not sure how to cancel out b to solve for a? 0313

 

[stapel]

Problem: F is an exponential function defined by f(x)= ab to the x power. In this problem, a and b are positive constants. If, f(5) is equal to 96 and f(7) is equal to 384, what is the value of a?

I am having a hard time conceptualizing this one, but (b/a) to the x power=96 and (b/a) to the x power equal to 384...

Which is it, "f(x) = a*b^x" or "f(x) = (b/a)^x"?

 

[soroban]

Hello, 0313phd!

\(\displaystyle f(x)\text{ is an exponential function defined by: }\,f(x)\:=\: ab^x,\,\text{ where }a,b\,>\,0.\)

\(\displaystyle \text{If }f(5)\,=\,96\,\text{ and }\,f(7)\,=\,384,\;\text{ find }a.\)

\(\displaystyle \text{We are given: }\;\begin{Bmatrix} f(5) &=& 96 & \Rightarrow & ab^5 &=& 96 & [1] \\ f(7) &=& 384 & \Rightarrow & av^7 &=& 384 & [2] \end{Bmatrix}\)

\(\displaystyle \text{Divide [2] by [1]: }\;\frac{ab^7}{ab^5} \:=\:\frac{384}{96} \quad\Rightarrow\quad b^2 \:=\:4 \quad\Rightarrow\quad b \:=\:2\)

\(\displaystyle \text{Substitute into [1]: }\;a(2^5) \:=\:96 \quad\Rightarrow\quad 32a \:=\:96 \quad\Rightarrow\quad \boxed{a \:=\:3}\)

 

[0313phd]

0313phd wrote:
YOU QUESTIONED, "Which is it, "f(x) = a*b^x" or "f(x) = (b/a)^x"? I reply," F is an exponential function defined by f(x)= ab to the x power. So it is a*b to the x power. a and b are positive constants. If, f(5) is equal to 96 and f(7) is equal to 384, what is the value of a?"

I just thought of something. Could you consider that you have two equations: one in which b is equal to 5, and the other in which b is equal to 7, so that: First equation: 5a to the x power= 96
Second equation: 7a to the x power= 384
But I still don't understand what the x power is, so I can't solve for a. Appreciate all your help. 0313

 

[0313phd]

Dear Soroban:
THANK YOU SO MUCH! 0313

 

[lookagain]

Problem: F is an exponential function defined by f(x)= ab to the x power.
In this problem, a and b are positive constants.

0313phd,

"ab to the x power" is not equal to \(\displaystyle ab^x.\)

It equals \(\displaystyle (ab)^x.\)


You could write \(\displaystyle ab^x \ as \ \ " a \ times \ the \ quantity \ of \ b \ raised \ to \ x."\)