Logarithm Questions
Logarithm questions with answers are provided for students to solve them and understand the concept elaborately. These questions are based on the logarithm chapter of Class 9, 10 and 11 syllabi. Practising these problems will not only help students to score good marks in academic exams but also participate in competitive exams conducted at the state or national level, such as Maths Olympiad.
The logarithmic function is an inverse of the exponential function. It is defined as:
y=log_{a}x, if and only if x=a^{y}; for x>0, a>0, and a≠1.
Natural logarithmic function: The log function with base e is called natural logarithmic function and is denoted by log_{e}.
f(x) = log_{e}x
The questions of logarithm could be solved based on the properties, given below:

Also, read:
Questions on Logarithm with Solutions
1. Express 5^{3 }= 125 in logarithm form.
Solution:
5^{3 }= 125
As we know,
a^{b} = c ⇒ log_{a}c=b
Therefore;
Log_{5}125 = 3
2. Express log_{10}1 = 0 in exponential form.
Solution:
Given, log_{10}1 = 0
By the rule, we know;
log_{a}c=b ⇒ a^{b} = c
Hence,
10^{0} = 1
3. Find the log of 32 to the base 4.
Solution: log_{4}32 = x
4^{x} = 32
(2^{2})^{x} = 2x2x2x2x2
2^{2x} = 2^{5}
2x=5
x=5/2
Therefore,
log_{4}32 =5/2
4. Find x if log_{5}(x7)=1.
Solution: Given,
log_{5}(x7)=1
Using logarithm rules, we can write;
5^{1} = x7
5 = x7
x=5+7
x=12
5. If log_{a}m=n, express a^{n1} in terms of a and m.
Solution:
log_{a}m=n
a^{n}=m
a^{n}/a=m/a
a^{n1}=m/a
6. Solve for x if log(x1)+log(x+1)=log_{2}1
Solution: log(x1)+log(x+1)=log_{2}1
log(x1)+log(x+1)=0
log[(x1)(x+1)]=0
Since, log 1 = 0
(x1)(x+1) = 1
x^{2}1=1
x^{2}=2
x=± √2
Since, log of negative number is not defined.
Therefore, x=√2
7. Express log(75/16)2log(5/9)+log(32/243) in terms of log 2 and log 3.
Solution: log(75/16)2log(5/9)+log(32/243)
Since, nlog_{a}m=log_{a}m^{n}
⇒log(75/16)log(5/9)^{2}+log(32/243)
⇒log(75/16)log(25/81)+log(32/243)
Since, log_{a}mlog_{a}n=log_{a}(m/n)
⇒log[(75/16)÷(25/81)]+log(32/243)
⇒log[(75/16)×(81/25)]+log(32/243)
⇒log(243/16)+log(32/243)
Since, log_{a}m+log_{a}n=log_{a}mn
⇒log(32/16)
⇒log2
8. Express 2logx+3logy=log a in logarithm free form.
Solution: 2logx+3logy=log a
logx^{2}+logy^{3}=log a [By logarithm rule: loga^{b} = b log a]
log(x^{2}y^{3})=log a [By logarithm rule: log a + log b = log (ab) ]
x^{2}y^{3} = a [If log_{m}a = log_{m}b, then a = b]
Video Lesson
Logarithmic Equations
9. Prove that: 2log(15/18)log(25/162)+log(4/9)=log2
Solution: 2log(15/18)log(25/162)+log(4/9)=log2
Taking L.H.S.:
⇒2log(15/18)log(25/162)+log(4/9)
⇒log(15/18)^{2}log(25/162)+log(4/9)
⇒log(225/324)log(25/162)+log(4/9)
⇒log[(225/324)(4/9)]log(25/162)
⇒log[(225/324)(4/9)]/(25/162)
⇒log(72/36)
⇒log2 (R.H.S)
10. Express log_{10}(2) + 1 in the form of log_{10}x.
Solution: log_{10}(2)+1
=log_{10}2+log_{10}10 [Since, 1 = log_{10}10 ]
=log_{10}(2 x 10) [log_{10} a + log_{10} b = log_{10} ab]
=log_{10}20
11. Find the value of x, if log_{10}(x10)=1.
Solution: Given, log_{10}(x10)=1.
log_{10}(x10) = log_{10}10
x10 = 10
x=10+10
x=20
12. Find the value of x, if log(x+5)+log(x5)=4log2+2log3
Solution: Given,
log(x+5)+log(x5)=4log2+2log3
log(x+5)(x5) = 4log2+2log3 [log mn=log m+log n]
log(x^{2}25) = log2^{4}+log3^{2}
log(x^{2}25) = log16+log9
log(x^{2}25)=log(16×9)
log(x^{2}25)=log144
x^{2}25=144
x^{2}=169
x=±√169
x=±13
13. Solve for x, if (log 225/log15) = log x
Solution: log x = (log 225/log15)
log x=[log(15×15)/log15]
log x = log 15^{2}/log 15
log x = 2log 15/log 15
log x = 2
Or
log_{10}x=2
10^{2}=x
x=10×10
x=100
Practice Questions
 If log x = m+n and log y=mn, express the value of log 10x/y^{2} in terms of m and n.
 Express 3^{2}=1/9 in logarithmic form.
 Express log_{10}0.01=2 in exponential form.
 Find the logarithm of 1/81 to the base 27.
 Find x if log_{7}(2x^{2}1)=2.
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