Question

16^log₄(5-√5)+4^log₂(√5+5)

Answer 1

$16^{log_4(5-\sqrt5)}+4^{log_2(\sqrt5+5)}=(4^2)^{log_4(5-\sqrt5)}+(2^2)^{log_2(\sqrt5+5)}=\\\\=4^{2log_4(5-\sqrt5)}+2^{2log_2(\sqrt5+5)}=4^{log_4(5-\sqrt5)^2}+2^{log_2(\sqrt5+5)^2}=\\\\=(5-\sqrt5)^2+(\sqrt5-5)^2=(25-10\sqrt5+5)+(5+10\sqrt5+25)=\\\\=30+30=60$

Answer 2

Смотри.....................

Answer 3

To solve the expression 16^log₄(5-√5) + 4^log₂(√5+5), we'll simplify each term step by step.

Step 1: Simplify the first term 16^log₄(5-√5) Let's rewrite 16 as 4^2 since 16 is 4 raised to the power of 2:

(4^2)^log₄(5-√5)

Now, remember that for any positive real numbers a, b, and c:

(a^b)^c = a^(b*c)

Using this property, we can simplify the expression further:

4^(2 * log₄(5-√5))

Step 2: Simplify the exponent 2 * log₄(5-√5) We can use the logarithmic property:

logₐ(b) = log_c(b) / log_c(a)

where "c" can be any base. Let's use "c" as 4:

log₄(5-√5) = log_c(5-√5) / log_c(4)

Now, evaluate the logarithms:

log₄(5-√5) = log(5-√5) / log(4)

Step 3: Substitute the result back into the expression:

4^(log(5-√5) / log(4))

Step 4: Simplify the second term 4^log₂(√5+5) Similarly, rewrite 4 as 2^2 since 4 is 2 raised to the power of 2:

(2^2)^log₂(√5+5)

Again, using the property (a^b)^c = a^(b*c):

2^(2 * log₂(√5+5))

Step 5: Simplify the exponent 2 * log₂(√5+5) As before, we'll use the logarithmic property:

log₂(√5+5) = log_c(√5+5) / log_c(2)

Let's evaluate the logarithms:

log₂(√5+5) = log(√5+5) / log(2)

Step 6: Substitute the result back into the expression:

2^(log(√5+5) / log(2))

Now, the entire expression becomes:

4^(log(5-√5) / log(4)) + 2^(log(√5+5) / log(2))

To get the numerical value, we need the actual values of log(5-√5), log(4), log(√5+5), and log(2). Let's calculate that:

log(5-√5) ≈ 0.399 log(4) = 2 (since log(4) = log₂(4) = 2) log(√5+5) ≈ 1.861 log(2) ≈ 0.301

Now, substitute these values into the expression:

4^(0.399 / 2) + 2^(1.861 / 0.301)

Calculate the individual terms:

4^(0.399 / 2) ≈ 4^0.1995 ≈ 1.414 2^(1.861 / 0.301) ≈ 2^6.1914 ≈ 64.077

Finally, add the two terms together:

1.414 + 64.077 ≈ 65.491

So, the value of the expression 16^log₄(5-√5) + 4^log₂(√5+5) is approximately 65.491.