Question

Сравнить: log2 5 + log2 3 и log2 (5+3)

Answer 1

Log2 5 + log2 3 = log2(5*3) = log2 15
log2 (5+3) = log2 8
log2 15 > log2 8
15>8

Answer 2

Так как основание логарифма 2>1, то логарифм здесь - возрастающая функция. Значит сравнивается по величине аргумент. То есть

$\log_2 a\ \textless \ \log_2 b$, если a,b>0 и a<b.

По свойству логарифмов

$\log_ca + \log_cb=\log_c(a*b)$

Здесь a,b,c>0 и с≠1.

$\log_25 + \log_23=\log_2(5*3)=\log_215$


$\log_2(5 + 3)=\log_28$.

Значит

$\log_28\ \textless \ \log_215$

Ответ: второе число меньше первого.

Answer 3

To compare the expressions, let's first simplify them:

1. log2 5 + log2 3: Using logarithm properties, we can combine the sum of logarithms into a single logarithm:

log2 5 + log2 3 = log2 (5 * 3) = log2 15

2. log2 (5 + 3): This is already a simplified expression.

Now we have log2 15 and log2 8. To compare them, we need to determine which one is larger.

Calculating the approximate values:

log2 15 ≈ 3.91 log2 8 = 3

So, log2 15 is greater than log2 8. Therefore, the comparison is as follows:

log2 5 + log2 3 > log2 (5 + 3) log2 15 > log2 8

Keep in mind that these are approximate values; the exact values of log2 15 and log2 8 extend infinitely beyond the decimal points.