If F(x) = 8^x contains the points A(a,3) and B(b,48), find the exact value of the slope of AB.
Could someone help me solve this please? It was on a practice SAT that I took, and I know I got it wrong cause I didn't know what to do.
Thanks
Hello,
the coordinates of the points have to satisfy the equation of the function:
to A: $\displaystyle 3=8^a \Longrightarrow a=\log_{8}{3}$
to B: $\displaystyle 48=8^b \Longrightarrow b=\log_{8}{48}$
The slope is calculated by: $\displaystyle m=\frac{y_B-y_A}{b-a}$ Therefore:
$\displaystyle m=\frac{48-3}{\log_{8}{48}-\log_{8}{3}}=\frac{45}{\log_{8}{\left( \frac{48}{3} \right)}}$=$\displaystyle \frac{45}{\frac{\ln{\left( \frac{48}{3} \right)}}{\ln(8)}}=\frac{45 \cdot \ln(8)}{\ln(16)}=\frac{45 \cdot 3\ln(2)}{4\ln(2)}=\frac{135}{4}$
EB