Question

1. the exponential function f can be written in the form $f(x) = a^x$. a table of selected values for f is shown.

x	f(x)
2.5	55.9
4.2	862.3
6	15624

a. find $\log_a(55.9)$

b. find the value of b is $\log_a(b) = 6$

Explanation:

Part (a)

Step1: Recall the definition of logarithm

By the definition of logarithms, if \( f(x)=a^x \), then \( \log_a(f(x)) = x \) because \( \log_a(y)=x \) is equivalent to \( a^x = y \).

Step2: Find the x - value for f(x) = 55.9

From the table, when \( f(x) = 55.9 \), the corresponding \( x \) - value is \( 2.5 \). So, by the definition of \( \log_a(y) \) (where \( y = f(x) \) and \( a^x=f(x) \)), we have \( \log_a(55.9)=2.5 \).

Step1: Recall the definition of logarithm

We know that if \( \log_a(b)=6 \), then by the definition of logarithms (since \( \log_a(y)=x \) is equivalent to \( a^x = y \)), we have \( a^6=b \).

Step2: Find the value of \( a^6 \) from the table

From the table, when \( x = 6 \), \( f(x)=a^6 = 15624 \). So, since \( \log_a(b)=6 \) implies \( b = a^6 \), we can conclude that \( b = 15624 \).

Answer:

\( 2.5 \)

Part (b)