Math, asked by Angel07082007, 4 months ago

Find the value of
log3 (log2 512)​

Answers

Answered by avniverma75
5

Step-by-step explanation:

Primeiramente, vamos lembrar da definição de logaritmo.

A definição de logaritmo nos diz que:

logₐ(b) = x ⇔ aˣ = b, sendo a > 0, a ≠ 1 e b > 0.

No logaritmo log₃(log₂(512)), vamos calcular o valor de log₂(512).

Igualando esse logaritmo a x, obtemos:

log₂(512) = x

2ˣ = 512.

Perceba que 512 é igual a 2⁹. Sendo assim:

2ˣ = 2⁹.

Como as bases são iguais, podemos igualar os expoentes. Assim, concluímos que x = 9.

Com isso, temos que log₃(log₂(512)) = log₃(9).

Vamos igualar esse logaritmo a x:

log₃(9) = x

3ˣ = 9.

Como 9 é igual a 3², então:

3ˣ = 3².

As bases são iguais, então o valor de x é:

x = 2.

Portanto, podemos afirmar que log₃(log₂(512)) é igual a 2.

Answered by Hansika4871
6

Given:

An expressionlog_{3}(log_{2}512).

To Find:

The value of the given expression.

Solution:

The given problem can be solved by using the properties of logarithms.

1. The given expression islog_{3}(log_{2}512).

2. According to the properties of logarithms,

=> log(a^b) = b (log a),

3. 512 can also be written as a power of 2,

=> 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, ( 2 repeated 9 times)

=> 512 = 2^9.

4. Substitute the value of 512 in power form in the given expression,

=> log_{3}(log_{2}2^9), ( log_{a}a =1),

=>log_{3}9(1),

=>log_{3}3^{2},

=> 2(log_{3}3),

=> 2 (1),

=> 2.

Therefore, the valuelog_{3}(log_{2}512) is 2.

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