Question

6*log1/2 32-20*log3 27+5*lg0,00001

Answer 1

Ответ:

1)

log3(27) - log(1/7)(7) = log3(3³) - log7(7)/log7(1/7) = 3log3(3) - 1/log7(7^(-1)) = 3 - 1/(-1) = 3 + 1 = 4.

2)

2^(1 + log2(5)) = 2^(log2(2) + log2(5)) = 2^log2(2 * 5) = 2^log2(10) = 10.

3)

lg4 + 2lg(5) = lg4 + lg(5²) = lg4 + lg(25) = lg(4 * 25) = lg100 = lg(10²) = 2.

4)

a)

3^x = 7;

log3(3^x) = log3(7);

xlog3(3) = log3(7);

x = log3(7).

b)

log4(x) = log0,5(√2);

log2(x)/log2(4) = log2(2^(1/2))/log2(1/2);

log2(x)/log2(2²) = (1/2)/log2(2^(-1/2));

log2(x)/2 = (1/2)/(-1/2);

log2(x)/2 = -1;

log2(x) = -2;

x = 2^(-2) = 1/4.

Answer 2

Let's break down and simplify the expression step by step:

1. 6 * log(1/2, 32):

   • log(1/2, 32) is the same as log2(32), as log(1/2, x) is equivalent to log2(x). And log2(32) means "what power do we raise 2 to in order to get 32?" which is 5.
   • So, 6 * log2(32) = 6 * 5 = 30.

2. 20 * log3(27):

   • log3(27) means "what power do we raise 3 to in order to get 27?" which is 3.
   • So, 20 * log3(27) = 20 * 3 = 60.

3. 5 * lg(0.00001):

   • lg(0.00001) is the same as log10(0.00001).
   • log10(0.00001) means "what power do we raise 10 to in order to get 0.00001?" which is -5.
   • So, 5 * log10(0.00001) = 5 * (-5) = -25.

Now, add up the results from the above steps:

30 - 60 - 25 = -55.

So, the simplified expression is -55.