Question

Порівняти log7(2) i log6(2)

Answer 1

$log_72=\frac{log_{10}2}{log_{10}7}=\frac{lg2}{lg7}\\\\log_62=\frac{lg2}{lg6}\\\\6$

Из дробей с одинаковыми числителями та меньше, знаменатель которой больше.

Answer 2

To compare the values of log7(2) and log6(2), we can use the property of logarithms that allows us to change the base of the logarithm. The change of base formula is as follows:

log_b(a) = log_c(a) / log_c(b)

where log_b(a) represents the logarithm of "a" with base "b," and log_c(x) is the logarithm of "x" with base "c."

Let's compare log7(2) and log6(2):

1. log7(2):

Using the change of base formula with base 10 (or any other consistent base), we have:

log7(2) = log(2) / log(7)

2. log6(2):

Similarly, using the change of base formula with base 10 (or any other consistent base):

log6(2) = log(2) / log(6)

Now, to compare the values, we need to determine whether log(2) / log(7) is greater than log(2) / log(6).

Since both expressions have the same numerator (log(2)), the ratio depends on the denominators (log(7) and log(6)). We don't need to calculate the actual numerical values of log(7) and log(6) to determine the comparison.

Here's what we know about logarithms:

1. log(x) increases as x increases.
2. The base of the logarithm is constant in this comparison (logarithms to the same base).

Since 7 is greater than 6, log(7) is greater than log(6). Therefore, log(2) / log(7) is less than log(2) / log(6).

In summary, we have:

log7(2) < log6(2)