Solve Logarithms Algebraically
This article will help you to solve logarithms algebraically.
The logarithm of a variety is the exponent by which another set value, the platform, has to be brought up to generate that variety. For example, the logarithm of 1000 to platform 10 is 3, because 1000 is 10 to the energy 3: 1000 = 10??10??10 = 103. More usually, if x = by, then y is the logarithm of x to platform b, and is published y = logb(x), so log10(1000) = 3.
Logarithms were presented by David Napier in the beginning Seventeenth millennium as a method for quickly simplify calculations. They were quickly implemented by navigators, researchers, technicians, and others to execute calculations more quickly, using fall guidelines and logarithm platforms. Boring multi-digit multiplication actions can be changed by desk look-ups and easier inclusion because of the truth essential in its own right that the logarithm of a item is the sum of the logarithms of the factors
loga mn = loga m + loga n [Product Law]
The above result is capable of extension i.e.
loga (mnp) = loga m + loga n + loga p +
loga m/n = loga m loga n [Quotient Law]
loga mn = n loga m [Power Law]
Example to Solve Logarithms Algebraically
Express log10 a2c/b in terms of log10 a, log10 b, log10 c.
Solution
log10 a2c/b = log10 a2c log10 b [Quotient Law]
= log10 a2 + log10 c log10 (b)1/2 [Product Law]
= 2 log10 a + log10 c 1/2 log10 b [Power Law]
Problems on Solve Logarithms Algebraically
Without using logarithm table, evaluate 3 + log10 (10 2)
Solution
3 + log10 (10 2) = 3 + (2) log10 [Power Law]
= 3 + (2). 1 [... Power Law]
= 3 2 = 1
Evaluate the following
(i) log 125/log 5
(ii) log672 log6 2
(iii) log4 8 log4 32
Solution
(i) log 125/log 5 = log 53/log 51/2 = 3 log 5/ 1/2 log 5 = 6
(ii) log6 72 log6 2 = log 72/2 = log6 36 = log6 62
= 2 log6 6 = 2 1 (... loga a = 1)
= 2
(iii) log4 8 log8 32 = log4 23 log8 25 = log4 (22)3/2 log8 (23)5/3
= log4 43/2 log8 85/3 = 3/2 log4 4 5/3 log8 8
= 3/2 1 5/3 1 (... loga a = 1)
= 3/2 5/3 = 9 10/6 = 1/6
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2.Express as a single logarithm : 2 + 1/2 log10 9 2 log10 5
Solution
2 + 1/2 log10 9 2 log10 5 = 2.1 + 1/2 log10 9 2 log10 5
= 2 log10 10 + log10 (9)1/2 log10 (5)2 [... log10 10 = 1]
= log10 (10)2 + log10 3 log10 25
= log (10)2 3/25 = log10 100 3/25 = log10 12
3.If log 7 log 2 + log 16 2 log 3 log 7/45 = 1 + log n, find n.
Solution
Given log 7 log 2 + log 16 2 log 3 log 7/45 = 1 + log n
log 7 log 2 + log 16 log (3)2 log 7/45 = log 10 + log n [... log 10 = 1]
log 7 16/2 32 7/45 = log (10 n) log 7 16 45/2 9 7 = log 10n
log 40 = log 10n 40 = 10n = 4
4. Solve the following 3x = 15
Solution:
Given exponential function is 3x = 15
Now we have to logarithmic on bath side
We get x log 3 = log 15
log 3 = 0.4771
log 10 = 1.176
x (0.4771) = 1.176
x =$frac{1.176}{0.4771}$ = 2.465
5.Solve for x: 3x = 8.
Solution: Take the logarithm of both
sides.
log 3x = log 8
Use theorem 2 to simplify the
equation.
x * log 3 = log 8
Solve for x by dividing
each side by log 3.
x = (log 8/log 3)
A decimal approximation may be
found if desired -
x = 1.8929.
by: mathqa
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