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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]?

1) 1.87
2) 2.25
3) 3.19
4) 3.72

Answer :

To find the value of [tex]\( b \)[/tex] in the exponential function [tex]\( f(x) = a(b)^x \)[/tex], we are given two points: [tex]\( f(5) = 15 \)[/tex] and [tex]\( f(8) = 170 \)[/tex].

### Step 1: Set Up the Equations
Using the exponential function formula:

1. For [tex]\( x = 5 \)[/tex]:
[tex]\[
a(b)^5 = 15
\][/tex]

2. For [tex]\( x = 8 \)[/tex]:
[tex]\[
a(b)^8 = 170
\][/tex]

### Step 2: Eliminate [tex]\( a \)[/tex]
To eliminate the variable [tex]\( a \)[/tex], divide the second equation by the first:

[tex]\[
\frac{a(b)^8}{a(b)^5} = \frac{170}{15}
\][/tex]

This simplifies to:

[tex]\[
(b)^{8-5} = \frac{170}{15}
\][/tex]

[tex]\[
b^3 = \frac{170}{15}
\][/tex]

### Step 3: Solve for [tex]\( b \)[/tex]
To find [tex]\( b \)[/tex], take the cube root of both sides:

[tex]\[
b = \left(\frac{170}{15}\right)^{1/3}
\][/tex]

After calculating, we find:

[tex]\[
b \approx 2.246
\][/tex]

### Conclusion
Comparing this value to the options provided:

- (1) 1.87
- (2) 2.25
- (3) 3.19
- (4) 3.72

The value of [tex]\( 2.246 \)[/tex] is closest to option (2) 2.25.

Therefore, the value of [tex]\( b \)[/tex] that is closest is option (2) 2.25.

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