A logarithm is a mathematical function designed to determine the exponent to which a particular base must be elevated to achieve a specified number. It stands as the reciprocal operation of exponentiation.
Logarithms represent the exponent to which a number must be raised to attain another specified number.
There are two types:
Logarithm with a base of 10 is referred to as the common logarithm and is expressed as log10 X. If the base is not specified, it is assumed to be 10.
(a) Natural Logarithm:
(b) Logarithm with a base of 'e' is known as the natural logarithm and is expressed as loge X.
Crucial Note: In the absence of a specified base, always assume the base to be 10
Value of log(2 to 10): Remember
Example 1: If log 27= 1.431, then the value of log 9 is?
(a) 0.945
(b) 0.934
(c) 0.958
(d) 0.954
Ans: (d)
log 27= 1.431
⇒ log(3)^{3}= 1.431
⇒ 3log 3= 1.431
⇒ log3= 0.477
therefore, log9= log 3^{2}= 2 log3= (2×0.477)= 0.954
Example 2: Solve the equation log x= 1 log(x3)
(a) 2
(b) 1/2
(c) 5
(d) 4
Ans: (c)
By combining both the equation we get
logx + log (x3)=1
log(x(x3))= log 10^{1}
Now convert it into exponential form,
x (x3)= 10^{1}
x^{2} – 3x10= 0
(x5) (x+2)=0
x= 2, x=5
By solving this equation we get two values for x.
x= 2, x=5
Put the different value of x in different equation and solve them,
x= 2
log(2) = 1 log (23)
x= 5
log5 = 1log(53)
log5 = 1log2
Negative value is not considered in logarithm. So, we have a single value of x i.e, x=5.
Example 3: If log _{10} 5+ log(5x+1) = log _{10} (x+5) +1, Find the value of X?
(a) 3
(b) 1
(c) 10
(d) 5
Ans: (a)
log _{10} 5+ log(5x+1) = log _{10} (x+5) +1.
log _{10} 5+ log(5x+1) = log _{10} (x+5) +log _{10 }10
log_{10} [5 (5x+1) ] = log_{10}( 10 (x+5)]
5 (5x+1) = 10 (x+5)
5x+1 = 2x+ 10
3x= 9
x=3.
Example 4: If log log(a+b), then
(a) ab=1
(b) a=b
(c) a+b=1
(d) a^{2}b^{2} = 1
Ans: (c)
so, a+b=1
Example 5: log_{9} (3log_{2} (1+log_{3} (1+2log_{2} x)))= 1/2. Find x.
(a) 2
(b) 1/2
(c) 1
(d) 4
Ans: (a)
_{}
log_{2}(1+log_{3}(1+2log_{2} x) = 1
1+log_{3} (1+2log_{2} x)= 2^{1}
log_{3} (1+2log_{2} x)=2^{1 }− 1
log_{3} (1+2log_{2} x) = 1
(1+2log_{2} x) = 3^{1}
1+ 2log_{2} x= 3
2log_{2} x = 2
log_{2}x = 1
x= 2
314 videos170 docs185 tests

1. What is a logarithm? 
2. What are the types of logarithms? 
3. What are some important formulas related to logarithms? 
4. How can logarithms be used in reallife applications? 
5. What are some frequently asked questions about logarithms in exams? 
314 videos170 docs185 tests


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