Suppose it is given that log_3 (2) = 0.63 and log_3 (34) = 3.2. Use these, and appropriate properties of logarithms, to find the values of the following logarithms.

Question

Part A: log_3 (36) Part B: log_3 (32) Part C: log_3 (1/17) Part D: log_3 (index 3√68)

Solve by using the given in the title.

Answer 1

In summary, when simplifying logarithmic expressions, two key things to looks for are:

1. Factoring the inner expression
2. Rewriting the inner expression in terms of an exponential (for example 32 can be thought of as 2⁵)

Part A.

log_3(36)

log_3(6²)

2*log_3(6)

2*log_3(2*3)

2(log_3(2) + log_3(3))

2(0.63 + 1)

3.26

Part B.

log_3(32)

log_3(2⁵)

5*log_3(2)

5*0.63

3.15

Part C.

log_3(1/17)

log_3(2/34)

log_3(2) - log_3(34)

0.63 - 3.2

-2.57

Part D.

log_3(index 3√68)

log_3(68^1/3) (third root can be rewritten as a fractional exponent)

(1/3)log_3(68)

(1/3)log_3(2*34)

(1/3)(log_3(2) + log_3(34))

(1/3)(0.63 + 3.2)

1.28

Answer 2

Using what is given as well as knowing the logarithmic rules and a little bit of thinking....

Given:

log₃ 2 = 0.63

log₃ 34 = 3.2

For the first problem, log₃ 36:

1. log₃ 36
2. log₃ ( 4 · 9 )
3. log₃ 4 + log₃ 9
4. log₃ 2² + log₃ 3²
5. 2 · log₃ 2 + 2 · log₃ 3
6. 2 ( 0.63 ) + 2 ( 1 )
7. 1.26 + 2
8. 3.26

For the second problem, log₃ 32:

1. log₃ 32
2. log₃ 2⁵
3. 5 · log₃ 2
4. 5 ( 0.63 )
5. 3.15

For the third problem, log₃ ( 1/17 ):

1. log₃ ( 1/17 )
2. log₃ 1 - log₃ 17
3. log₃ 1 - log₃ ( 34 / 2 )
4. log₃ 1 - ( log₃ 34 - log₃ 2 )
5. log₃ 1 - log₃ 34 + log₃ 2
6. 0 - 3.2 + 0.63
7. -2.57

For the fourth problem, log₃ ³√68:

1. log₃ ³√68
2. log₃ ( 68 )^1/3
3. 1/3 · log₃ 68
4. 1/3 · log₃ ( 2 · 34 )
5. 1/3 ( log₃ 2 + log₃ 34 )
6. 1/3 ( 0.63 + 3.2 )
7. 1/3 ( 3.83 )
8. ≈ 1.27666666
9. ≈ 1.3