Change of Base Formula

The values of log 2 and log 7 are given.

So change the base to 10.

So log₂ 70 = [log 70]/[log 2].

Common Logarithm

70 = 7⋅10

log 7⋅10 = log 7 + log 10

Logarithm of a Product

It says
assume log 2 = 0.301, log 7 = 0.845.

So
[log 7 + log 10]/[log 2]
= [0.845 + 1]/[0.301].

0.845 + 1 = 1.845

1.845/0.301 = 1845/301

1845/301 = 6.129...

Round this to the nearest hundreadth.

(0.301, 0.845 has 3 significant digits.
So round 6.129... to make 3 significant digits.)

Then 1845/301 = 6.13.

So 6.13 is the answer.

log₂ 3 = a is given.

So change the base to 2.

log₁₂ 18 = [log₂ 18]/[log₂ 12]

Write the prime factorizations of 18 and 12.

18 = 2⋅3²
12 = 2²⋅3

log₂ (2⋅3²) = log₂ 2 + log₂ 3²

log₂ (2²⋅3) = log₂ 2² + log₂ 3

+log₂ 3² = +2 log₂ 3

log₂ 2² = 2 log₂ 2

log₂ 2 = 1

log₂ 3 = a

1 + 2a = 2a + 1

2⋅1 + a = a + 2

So
[2a + 1]/[a + 2]
is the answer.

Change the bases of the logs to 2.

You don't have to change log₂ 27.

log₉ 16 = [log₂ 16]/[log₂ 9].

You can also change the bases to 3.
You'll get the same answer.

27 = 3³

16 = 2⁴

9 = 3²

Power

log₂ 3³ = 3 log₂ 3

log₂ 2⁴ = 4 log₂ 2

log₂ 3² = 2 log₂ 3

Cancel (log₂ 3).

log₂ 2 = 1

4⋅1/2 = 2

3⋅2 = 6

So 6 is the answer.