SME
Grade 11 Math · Problem type
Common logs from scientific notation
Use this for “Find log 3,000 using log 3 ≈ 0.4771.”
Diagram
3,000 = 3 × 10³, so its log is 3 plus the log of 3.
log₁₀ 3,000 = 3 + log₁₀ 3 ≈ 3 + 0.477 = 3.4771
The power of ten gives the whole part; the ruler gives the decimal part.
This is an example. Type your own values to start a new problem.
N
=
a
×
10
log
N
=
n
+
log
a
=
L
Formulas
N
=
a
×
10
n
3,000
=
3 ×
10
3
n
=
exponent
of
the
power
of
ten
at
or
below
N
3 = exponent of the power of ten at or below 3,000
L
=
n
+
log₁₀
a
3.4771 = 3 + log₁₀ 3
Assumptions
•
Write N as a × 10ⁿ with a from 1 up to 10.
•
log(a × 10ⁿ) = n + log a: the exponent is the whole part, and log a, from 0 up to 1, is the rest.
•
A number under 1 has a negative n, so its log is negative.
Step-by-step
WE KNOW
N = 3,000
FIND
power of ten (n), number from 1 up to 10 (a), log₁₀ N (L)
Step 1 · Find power of ten (n)
Use n = exponent of the power of ten at or below N
How many places the point moves: the whole part of the log.
n
=
exponent
of
the
power
of
ten
at
or
below
3,000
n
=
3
Step 2 · Find number from 1 up to 10 (a)
Use N = a × 10ⁿ
Divide by the power of ten to leave one digit, not 0, before the point.
a
=
N
10
n
a
=
3,000
10
3
a
=
3,000
1,000
a
=
3
Step 3 · Find log₁₀ N (L)
Use L = n + log₁₀ a
The
log
of
a
product
is
the
sum
of
the
logs,
and
log₁₀
10ⁿ
=
n
.
L
=
3
+
log₁₀
3
L
=
3
+
0.4771
L
=
3.4771
Check
Put the numbers back in. Both sides should match.
3,000
=
3 ×
10
3
✓
3 = exponent of the power of ten at or below 3,000 ✓
3.4771 ≈ 3 + log₁₀ 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.