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In an exponential function, [tex]f(x) = a(b)^x[/tex], it is known that [tex]f(5) = 15[/tex] and [tex]f(8) = 170[/tex]. Which of the following is closest to the value of [tex]b[/tex]?

Answer :

To solve for the value of [tex]\( b \)[/tex] in the exponential function [tex]\( f(x) = a(b)^x \)[/tex], given that [tex]\( f(5) = 15 \)[/tex] and [tex]\( f(8) = 170 \)[/tex], we can follow these steps:

1. Set Up the Equations:
- From [tex]\( f(5) = 15 \)[/tex], we have:
[tex]\[
a(b)^5 = 15
\][/tex]
- From [tex]\( f(8) = 170 \)[/tex], we have:
[tex]\[
a(b)^8 = 170
\][/tex]

2. Eliminate [tex]\( a \)[/tex]:
- Divide the second equation by the first equation to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a(b)^8}{a(b)^5} = \frac{170}{15}
\][/tex]
- Simplifying the left side:
[tex]\[
(b)^8 / (b)^5 = b^{8-5} = b^3
\][/tex]
- Therefore:
[tex]\[
b^3 = \frac{170}{15}
\][/tex]

3. Calculate [tex]\( b^3 \)[/tex]:
- Compute the division:
[tex]\[
b^3 = \frac{170}{15} \approx 11.3333333333
\][/tex]

4. Solve for [tex]\( b \)[/tex]:
- To find [tex]\( b \)[/tex], take the cube root of [tex]\( b^3 \)[/tex]:
[tex]\[
b = \sqrt[3]{11.3333333333} \approx 2.2462213669
\][/tex]

Therefore, the value of [tex]\( b \)[/tex] that is closest is approximately [tex]\( 2.25 \)[/tex].

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