CONTENTS:
|
1 |
Definition |
|
2 |
Formulas |
|
3 |
19 Selected Problems |
|
4 |
Solutions to the Selected Problems |
Definition
A logarithm is a mathematical function that expresses how many times a base number must be multiplied by itself to reach a given number. In simpler terms, it's the inverse operation of exponentiation. For example, if 10 is raised to the power of 3 (10^3), the result is 1000. The logarithm (base 10) of 1000 is 3, because 10 must be multiplied by itself 3 times to get 1000.
Mathematically,
|
If
ax = N (where, a > 0 but a ≠ 1), |
|
logaN
= x [Read as logarithm of ‘N’ to the base ‘a’ is equal to ‘x’] |
Formulas
|
1 |
loga1
= |
0 |
|
2 |
logaa
= |
1 |
|
3 |
loga(m
× n) = |
logam
+ logan |
|
4 |
loga(m
÷ n) = |
logam
− logan |
|
5 |
logamn
= |
n
× logam |
|
6 |
logam
= |
logbm
× logab |
|
7 |
logab
× logba = |
1 |
|
8 |
logam
= |
logbm
÷ logba |
|
9 |
e^(logey)
= |
y |
Selected
Problems
|
1 |
Find the value of: log264 |
|
2 |
Express with base value as 3: log23 |
|
3 |
Express with base value as 3 after simplification: log123 |
|
4 |
Find the value of: 3^(log39) |
|
5 |
Find the logarithm of 2025 to the base 3√5 |
|
6 |
The logarithm of a number to the base √2 is k. What is its logarithm to
the base 2√2? |
|
7 |
If log2x + log4x + log16x = 21/4, find
x. |
|
8 |
If p = log1020 and q = log1025, find the value of
‘x’ such that 2log10(x+1) = 2p−q |
|
9 |
If x = log2aa, y = log3a2a and z = log4a3a,
show that xyz + 1 = 2yz |
|
10 |
Show that log3√3√3√3........∞ = 1 |
|
11 |
Prove that [1 ÷ loga(ab)] + [1 ÷ logb(ab)] = 1 |
|
12 |
Find logarithm of 0.0001 to the base 0.01 |
|
13 |
Find logarithm of 0.333 to the base 81 |
|
14 |
Log (a9) + log (a) = 10. Find ‘a’ |
|
15 |
Find the value of log2log2log216 |
|
16 |
Find the value of logab + log(a^2)(b^2) + log(a^3)(b^3)
+...........+ log(a^n)(b^n) |
|
17 |
If log t + log (t – 3) = 1, find ‘t’ |
|
18 |
Find the value of logba×logcb×logac |
|
19 |
Find the value of log(a2/bc) + log(b2/ca) + log(c2/ab) |
Logarithm
Solutions
to the Selected Problems
