Question

Using the change formula, determine which of the following logarithms has the same value as log_64 (1/16). (1 point)

1)log_1/2 (1/8)
2)log_27 (1/9)
3)log_36 (1/216)
4)log_16?(1/64)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine which of the given options has the same value as log_64 (1/16), we can use the change of base formula which states:

log_a (b) = log_c (b) / log_c (a)

Given log_64 (1/16), we can rewrite it using the change of base formula by converting it to base 2 logarithm, as follows:

log_64 (1/16) = log_2 (1/16) / log_2 (64)

Next, simplify the numerator and the denominator:

log_64 (1/16) = log_2 (1/16) / log_2 (2^6)
log_64 (1/16) = log_2 (1/16) / 6

log_64 (1/16) = log_2 (1/2^4) / 6
log_64 (1/16) = log_2 (1/2)^4 / 6
log_64 (1/16) = 4 / 6
log_64 (1/16) = 2 / 3

Now, let's check the options:

1) log_1/2 (1/8) = 3 / 3
2) log_27 (1/9) = 2 / 3
3) log_36 (1/216) = 3 / 3
4) log_16 (1/64) = 2 / 3

The option 2) log_27 (1/9) has the same value as the given log_64 (1/16).