## Rules of logarithms

Rules of logarithms

1. Product formula:

loga(mn)=logam + logan

2. Quotient formula:

loga(m/n)=logam - logan.

3. Power formula:

logamn= nlogam.

4. Change of base formula:

lognm = logam + logn a.

• logm n = 1/(lognm)
• If a is any positive number, then logm 1 =0

• If a is any positive number, then logaa=1

• If m, n and a be positive numbers and a ≠1, and if loga m = logan, then m = n.
• If a and b are any two positive numbers and b ≠, then b logab a = a.

Simplify:

log3 27 + log3 729

Solution:

log3 27 + log3 729

We know that 27 = 33 and 729 = 36

log3 27 + log3 729=log3 (27x729)[by product formula]

= log 3 (33x36)

= log 3 33+6

= log 3 39

=9log3 3[by power rule]

=9(1)[by loga a =1]

= 9

Solve:

log2 (7x + 3) - log2(5x - 1)= log23 - 1

Solution:

log2 (7x + 3)-log2(5x - 1)= log23 - 1

log2 [(7x + 3)/(5x - 1)] = log23 - log22

log2 [(7x + 3)/(5x - 1)] = log2(3/2)

[(7x + 3)/(5x - 1)] = 3/2

2(7x + 3) = 3(5x-1)

14x + 6 = 15x - 3

6 + 3 = 15x -14x

9 = x

x = 9

Simplify:

3log62 - log648

Solution:

log62 3 - log648

log68 - log648

= log6 (8/48)

= log6 (1/6)

= log6 6-1

= -1 log6 6

= -1 x 1

-1

Rules of logarithms to Home page

Logarithms

Math dictionary