Find logarithm of 32: We need to find the value of the logarithm of 32 with base 1/2. This means we are looking for the exponent that we need to raise 1/2 to in order to get 32.Express 32 as a power of 2: We can rewrite 32 as a power of 2, since 32 is 2 raised to the 5th power: 32 = 2^5.Convert logarithm to base 2: Now, we can express the logarithm in terms of base 2: log_(1/2) 32 is the same as asking ＂to what power must we raise 1/2 to get 2^5?＂ Since raising a number to a negative exponent is the same as taking the reciprocal of the base raised to the positive exponent, we can write this as log_(1/2) 2^5 = -log_2 2^5.Evaluate logarithm of 2^5: We know that log_2 2^5 = 5 because the base and the argument are the same, and the exponent is the answer to the logarithm.Final result: Therefore, log_(1/2) 32 = -log_2 2^5 = -5.

Grade 8

Relationship between squares and square roots

Full solution

\log _{\frac{1}{2}} 32=

Find logarithm of $32$ : We need to find the value of the logarithm of $32$ with base $\frac{1}{2}$ . This means we are looking for the exponent that we need to raise $\frac{1}{2}$ to in order to get $32$ .
Express $32$ as a power of $2$ : We can rewrite $32$ as a power of $2$ , since $32$ is $2$ raised to the $5$ th power: $32 = 2^5$ .
Convert logarithm to base $2$ : Now, we can express the logarithm in terms of base $2$ : $\log_{\frac{1}{2}} 32$ is the same as asking ＂to what power must we raise $\frac{1}{2}$ to get $2^5$ ?＂ Since raising a number to a negative exponent is the same as taking the reciprocal of the base raised to the positive exponent, we can write this as $\log_{\frac{1}{2}} 2^5 = -\log_2 2^5$ .
Evaluate logarithm of $2^5$ : We know that $\log_2 2^5 = 5$ because the base and the argument are the same, and the exponent is the answer to the logarithm.
Final result: Therefore, $\log_{\frac{1}{2}} 32 = -\log_2 2^5 = -5$ .