SME
Grade 11 Math · Problem type
Change of base
Use this for “Estimate log₃ 20” with the log key on a calculator.
Chart
Zero: x = 1
Vertical asymptote x = 0
f(20) ≈ 2.7268
Common log of x, log x: 1.301
Common log of b, log b: 0.4771
This is an example. Type your own values to start a new problem.
log
(
x
)
=
log
log
=
y
L₁
Common log of x, log x
worked out (not typed)
L₂
Common log of b, log b
worked out (not typed)
Formulas
L₁
=
log₁₀
x
1.301 = log₁₀ 20
L₂
=
log₁₀
b
0.4771 = log₁₀ 3
y
=
L₁
L₂
2.7268
=
1.301
0.4771
Assumptions
•
logb(x) = log x ÷ log b, with the common logs (base 10) on a calculator.
•
Any base works for both logs, if both use the same one.
•
The base b is positive and not 1, and x is positive.
Step-by-step
WE KNOW
b = 3, x = 20
FIND
common log of x, log x (L₁), common log of b, log b (L₂), logb(x) (y)
Step 1 · Find common log of x, log x (L₁)
Use L₁ = log₁₀ x
The
common
log:
the
power
of
10
that
makes
x
.
L₁
=
log₁₀
20
L₁
=
1.30103
Step 2 · Find common log of b, log b (L₂)
Use L₂ = log₁₀ b
The common log of the base.
L₂
=
log₁₀
3
L₂
=
0.47712
Step 3 · Find logb(x): y
Use y = L₁ ÷ L₂
Change of base: divide the two common logs.
y
=
1.30103
0.47712
y
=
2.7268
Check
Put the numbers back in. Both sides should match.
1.30103 ≈ log₁₀ 20 ✓
0.47712 ≈ log₁₀ 3 ✓
2.7268
≈
1.30103
0.47712
✓
Topic check
See what you know: 5 questions on this topic
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