SME
Engineering Programming (MATLAB/Python) · Problem type
Exponential rate from two points on a semilog plot
Use this for “Data points (0, 5) and (4, 20) lie on a straight line on semilog axes. Find k in y = y₀eᵏˣ.”
Chart
y(x) = 5 e0.3466x
A log y-axis: an exponential draws as a straight line.
This is an example. Type your own values to start a new problem.
y₁
=
y₀
e
×
y₂
=
y₀
e
×
Formulas
k
=
ln(
y₂
÷
y₁
)
x₂
−
x₁
0.3466
=
ln(20 ÷ 5)
4 − 0
y₁
=
y₀
×
e
k
×
x₁
5
=
5
×
e
0.3466 × 0
y₂
=
y₀
×
e
k
×
x₂
20
=
5
×
e
0.3466 × 4
Assumptions
•
The data follow y = y₀ekx, so ln y = ln y₀ + kx: a straight line with a log y-axis.
•
Both y values are above 0; a log axis has no place for 0 or a negative.
•
k > 0 is growth and k < 0 is decay; y₀ is y at x = 0.
Step-by-step
WE KNOW
x₁ = 0, y₁ = 5, x₂ = 4, y₂ = 20
FIND
rate (k), starting value (y₀)
Step 1 · Find rate (k)
Use k = ln(y₂ ÷ y₁) ÷ (x₂ − x₁)
With a log y-axis the slope is the rise in ln y over the run in x.
k
=
ln(20 ÷ 5)
4 − 0
k
=
ln(4)
4
k
=
1.3863
4
k
=
0.3466
Step 2 · Find starting value (y₀)
Use y₁ = y₀ × ek × x₁
Put
the
first
point
into
y
=
y₀
e
kx
and
multiply
by
e
−
kx₁
.
y₀
=
y₁
×
e
−
k
×
x₁
y₀
=
5
×
e
−0.3466 × 0
y₀
=
5
×
e
0
y₀
=
5
Check
Put the numbers back in. Both sides should match.
0.3466
=
ln(20 ÷ 5)
4 − 0
✓
5
=
5
×
e
0.3466 × 0
✓
20
=
5
×
e
0.3466 × 4
✓
Topic check
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