Question

2.15
dg plots
a semi - log plot:
→ scales the y - axis logarithmically.
→ exponential data or functions appear linear.
given the exponential function ( f(x)=2cdot7^{x} ). complete the table of values. graph the points on the semi - log plot. what do you notice about the points on the semi - log plot? (hint: ( 7^{3}=243 ) and ( 7^{4}=2401 ))

( x )	( f(x) )
1
2
3
4

Explanation:

Step1: Calculate \( f(x) \) for \( x = 0 \)

The function is \( f(x)=2\cdot7^{x} \). Substitute \( x = 0 \):
\( f(0)=2\cdot7^{0}=2\cdot1 = 2 \)

Step2: Calculate \( f(x) \) for \( x = 1 \)

Substitute \( x = 1 \) into \( f(x)=2\cdot7^{x} \):
\( f(1)=2\cdot7^{1}=2\cdot7 = 14 \)

Step3: Calculate \( f(x) \) for \( x = 2 \)

Substitute \( x = 2 \):
\( f(2)=2\cdot7^{2}=2\cdot49 = 98 \)

Step4: Calculate \( f(x) \) for \( x = 3 \)

Substitute \( x = 3 \). Given \( 7^{3}=343 \) (note: the hint had a typo, \( 7^{3}=343 \), not 243), so:
\( f(3)=2\cdot343 = 686 \)

Step5: Calculate \( f(x) \) for \( x = 4 \)

Substitute \( x = 4 \). Given \( 7^{4}=2401 \), so:
\( f(4)=2\cdot2401 = 4802 \)

Answer:

The completed table is:

\( x \)	\( f(x) \)
1	14
2	98
3	686
4	4802

For the semi - log plot (logarithmic y - axis), when we plot the points \( (x,\log(f(x))) \) (or \( (x,f(x)) \) with y - axis logarithmic), the points will lie on a straight line because for an exponential function \( y = ab^{x} \), taking the logarithm (base 10 or natural) of both sides gives \( \log(y)=\log(a)+x\log(b) \), which is a linear equation in the form \( Y = mX + c \) where \( Y=\log(y) \), \( X = x \), \( m=\log(b) \) and \( c=\log(a) \). So the points on the semi - log plot are collinear (lie on a straight line).