Science > Mathematics > Algebra > Logarithms > Use of the Laws of Logarithms Set – II
In the last article, we have studied to solve problems on the laws of logarithms to evaluate or simplify given logarithmic expression. In this article, we shall study to solve more problems on these laws to prove given logarithmic expression.
Laws of Logarithms
- Log a + Log b = Log (ab) (Law of Product)
- Log a – log b = log (a/b) b ≠ 0 (law of Quotient)
- Log am = m Log a (Law of exponent)
- Log (1) = 0
- Logaa = 1
Example 01:
Show that,
log 360 = 3 log2 + 2 log 3+ log 5
Solution:
L.H.S. = log 360
∴ L.H.S. = log (2 x 2 x 2 x 3 x 3 x 5)
∴ L.H.S. = log (23 x 32 x 5)
∴L.H.S. = log 23 + log 32 + log 5
∴L.H.S. = 3 log2 + 2 log 3+ log 5
∴ L.H.S. = R.H.S.
∴Log 360 = 3 log2 + 2 log 3+ log 5 (Proved as required)
Example 02:
Show that,
log 540 = 2 log2 + 3 log 3+ log 5
Solution:
L.H.S. = log 540
∴ L.H.S. = log (2 x 2 x 3 x 3 x 3 x 5)
∴ L.H.S. = log (22 x 33 x 5)
∴L.H.S. = log 22 + log 33 + log 5
∴L.H.S. = 2 log2 + 3 log 3+ log 5
∴ L.H.S. = R.H.S.
∴Log 540 = 2 log2 + 3 log 3+ log 5 (Proved as required)
Example 03:
Show that
Solution:
(Proved as required)
Example 04:
Show that
Solution:
(Proved as required)
Example 05:
Show that
Solution:
(Proved as required)
Example 06:
Show that
Solution:
(Proved as required)
Example 07:
Show that
Solution:
(Proved as required)
Example 08:
Show that
Solution:
Example 09:
Show that
Log(log x7) – log(logx3) = log(7/3)
Solution:
Log(log x7) – log(logx3) = log(7/3)
(Proved as required)
Example 10:
Show that
Log(log x4) – log(logx) = log 4
Solution:
Log(log x4) – log(log x) = log 4
(Proved as required)
Example 11:
Show that
Log(log x4) – log(logx6) = log(2/3)
Solution:
Log(log x4) – log(logx6) = log(2/3)
(Proved as required)
Example 12:
Show that
Solution:
(Proved as required)
Example 13:
Show that
Solution:
(Proved as required)
Example 14:
Show that
log(1 + 2 + 3) = log 1 + log2 + log3
Solution:
L.H.S. = log(1 + 2 + 3)
∴ L.H.S. = log(6)
∴ L.H.S. = log(1 x 2 x 3)
∴ L.H.S. = log 1 + log2 + log3
∴ L.H.S. = R.H.S.
∴ log(1 + 2 + 3) = log 1 + log2 + log3
(Proved as required)
Example 15:
Show that
Solution:
(Proved as required)
In the next article, we shall study to solve more problems on these laws to prove given relation using given logarithmic expression.