For AI agents: use /llms.txt for the Nakafa content index.
Like exponents, logarithms also have several important properties that need to be understood. These properties will be very helpful in solving various logarithmic problems.
Let a > 0 a > 0 a > 0 a ≠ 1 a \neq 1 a = 1
a , b , c , m , n a, b, c, m, n a , b , c , m , n ( a , b , c , m , n ∈ R ) (a, b, c, m, n \in \mathbb{R}) ( a , b , c , m , n ∈ R ) logarithm properties
a log a = 1 ^a\log a = 1 a log a = 1
a log 1 = 0 ^a\log 1 = 0 a log 1 = 0
a log a n = n ^a\log a^n = n a log a n = n
a log ( b × c ) = a log b + a log c ^a\log (b \times c) = ^a\log b + ^a\log c a log ( b × c ) = a log b + a log c
a log ( b c ) = a log b − a log c ^a\log \left(\frac{b}{c}\right) = ^a\log b - ^a\log c a log ( c b ) = a log b − a log c
a log b n = n ⋅ a log b ^a\log b^n = n \cdot ^a\log b a log b n = n ⋅ a log b
a log b = m log b m log a = 1 b log a ^a\log b = \frac{^m\log b}{^m\log a} = \frac{1}{^b\log a} a log b = m l o g a m l o g b = b l o g a 1
a log b × b log c = a log c ^a\log b \times ^b\log c = ^a\log c a log b × b log c = a log c
Property 4 4 4 a log ( b × c ) = a log b + a log c ^a\log (b \times c) = ^a\log b + ^a\log c a log ( b × c ) = a log b + a log c
Proof: Let a log b = m ^a\log b = m a log b = m a log c = n ^a\log c = n a log c = n
b = a m b = a^m b = a m c = a n c = a^n c = a n
Using the property of exponents:
Property 5 5 5 a log ( b c ) = a log b − a log c ^a\log \left(\frac{b}{c}\right) = ^a\log b - ^a\log c a log ( c b ) = a log b − a log c
Proof: Let a log b = m ^a\log b = m a log b = m a log c = n ^a\log c = n a log c = n
Then b = a m b = a^m b = a m c = a n c = a^n c = a n
Recall that a m a n = a m − n \frac{a^m}{a^n} = a^{m-n} a n a m = a m − n
Property 6 6 6 a log b n = n ⋅ a log b ^a\log b^n = n \cdot ^a\log b a log b n = n ⋅ a log b
Proof: Let a log b = m ^a\log b = m a log b = m
a log b n ^a\log b^n a log b n b b b n n n
a log b n = a log ( b × b × b × … × b ⏟ n factors ) ^a\log b^n = ^a\log (\underbrace{b \times b \times b \times \ldots \times b}_{n \text{ factors}}) a log b n = a log ( n factors b × b × b × … × b ) Using property 4 4 4
Property 7 7 7 a log b = m log b m log a = 1 b log a ^a\log b = \frac{^m\log b}{^m\log a} = \frac{1}{^b\log a} a log b = m l o g a m l o g b = b l o g a 1
Proof:
Based on the definition of logarithm, a log b = c ^a\log b = c a log b = c b = a c b = a^c b = a c
Suppose we use base m m m b b b
Since c = a log b c = ^a\log b c = a log b
Property 8 8 8 a log b × b log c = a log c ^a\log b \times ^b\log c = ^a\log c a log b × b log c = a log c
Proof:
Based on the definition:
Substitute the value of b b b c c c
Since c = a m n c = a^{mn} c = a mn
Suppose we want to calculate 5 log 125 ^5\log 125 5 log 125
Simplify the following expressions:
9 log 81 ^9\log 81 9 log 81
2 log 64 − 2 log 16 ^2\log 64 - ^2\log 16 2 log 64 − 2 log 16
4 log 16 10 ^4\log 16^{10} 4 log 1 6 10
If 5 log 4 = m ^5\log 4 = m 5 log 4 = m 4 log 3 = n ^4\log 3 = n 4 log 3 = n 12 log 100 ^{12}\log 100 12 log 100
The population of city A A A 2010 2010 2010 300,000 people 300{,}000 \text{ people} 300 , 000 people 6 % 6\% 6% A A A 1 million 1 \text{ million} 1 million
How much time is needed for Dini's money, which was initially Rp 2,000,000.00 \text{Rp}2{,}000{,}000.00 Rp 2 , 000 , 000.00 Rp 6,500,000.00 \text{Rp}6{,}500{,}000.00 Rp 6 , 500 , 000.00 12 % 12\% 12%
Determining logarithm values
Answer:
Answer:
Answer:
Given that 5 log 4 = m ^5\log 4 = m 5 log 4 = m 4 log 3 = n ^4\log 3 = n 4 log 3 = n
Then:
The initial population is 300,000 people 300{,}000 \text{ people} 300 , 000 people
The annual population growth is 6 % 6\% 6%
The appropriate function to describe population growth in x years x \text{ years} x years
The initial savings are Rp 2,000,000.00 \text{Rp}2{,}000{,}000.00 Rp 2 , 000 , 000.00
The final savings are Rp 6,500,000.00 \text{Rp}6{,}500{,}000.00 Rp 6 , 500 , 000.00
The interest rate is 12 % 12\% 12%
The appropriate function to describe Dini's savings in x years x \text{ years} x years
For a population of 1,000,000 people 1{,}000{,}000 \text{ people} 1 , 000 , 000 people
Therefore, the population will reach 1,000,000 people 1{,}000{,}000 \text{ people} 1 , 000 , 000 people 20 20 20 21 years 21 \text{ years} 21 years
For a final saving amount of Rp 6,500,000.00 \text{Rp}6{,}500{,}000.00 Rp 6 , 500 , 000.00
Therefore, Dini's savings will reach Rp 6,500,000.00 \text{Rp}6{,}500{,}000.00 Rp 6 , 500 , 000.00 10 years 10 \text{ years} 10 years
