### Video Transcript

What is the value of log 128 to the base one-half?

We recall that logarithms are the opposite or inverse of exponentials. If log of 𝑏 to the base 𝑎 is equal to 𝑐, then 𝑎 to the power of 𝑐 is equal to 𝑏. In this question, our value of 𝑎 is one-half or 0.5. Our value of 𝑏 is 128. And we need to calculate the value of 𝑐.

Substituting in these values, we have one-half to the power of 𝑐 equals 128, where we need to calculate the exponent or power 𝑐. We know that a negative exponent means the reciprocal. For example, 𝑥 to the power of negative one is equal to one over 𝑥. This means that one-half to the power of negative one is equal to two, as two is the reciprocal of one-half. Two to the power of seven is equal to 128. This means that one-half to the power of negative seven will also be 128.

The exponent 𝑐 is equal to negative seven. We can therefore conclude that log of 128 to the base one-half is equal to negative seven.