Question

worksheet a: (topic 2.14) logarithmic modeling
name:

1. selected values of the logarithmic function ( f ) are given in the table above, where ( f(x) = a + b ln x ).

a) use the data to write two equations that can be used to find the values for constants ( a ) and ( b ) in the expression for ( f(x) ).
b) find the values of ( a ) and ( b ).

the table has two rows and three columns. the first row (x) has values 1, 6. the second row (f(x)) has values 2, 10.

Explanation:

Part (a)

Step1: Substitute \( x = 1, f(x)=2 \)

We know the function is \( f(x)=a + b\ln x \). Substitute \( x = 1 \) and \( f(x)=2 \) into the function. Since \( \ln 1=0 \), we get \( 2=a + b\ln 1 \), which simplifies to \( 2=a + b\times0 \), so \( 2=a \).

Step2: Substitute \( x = 6, f(x)=10 \) and \( a = 2 \)

Now substitute \( x = 6 \), \( f(x)=10 \) and \( a = 2 \) into \( f(x)=a + b\ln x \). We get \( 10=2 + b\ln 6 \).

Step1: Use the first equation to find \( a \)

From part (a), we have the equation \( 2=a + b\ln 1 \). Since \( \ln 1 = 0 \), this simplifies to \( a=2 \).

Step2: Substitute \( a = 2 \) into the second equation

We have the second equation \( 10=2 + b\ln 6 \). Subtract 2 from both sides: \( 10 - 2=b\ln 6 \), so \( 8=b\ln 6 \). Then solve for \( b \): \( b=\frac{8}{\ln 6} \).

Answer:

The two equations are \( 2=a + b\ln 1 \) (or \( 2 = a \)) and \( 10=2 + b\ln 6 \)

Part (b)