Respuesta :
The value of the logarithm expression [tex]log_{0.5}(16)[/tex] evaluated is given by : Option A: -4.00
What is logarithm and some of its useful properties?
When you raise a number with an exponent, there comes a result.
Lets say you get
[tex]a^b = c[/tex]
Then, you can write 'b' in terms of 'a' and 'c' using logarithm as follows
[tex]b = log_a(c)[/tex]
Some properties of logarithm are:
[tex]\log_a(b) = \log_a(c) \implies b = c\\\\\log_a(b) + \log_a(c) = \log_a(b \times c)\\\\\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})\\\\\log_a(b^c) = c \times \log_a(b)\\\\\log_b(b) = 1\\\\\log_a(b) + log_b(c) = \log_a(c)\\[/tex]
Log with base e = 2.71828... is written as [tex]\ln(x)[/tex] simply.
Log with base 10 is written as [tex]\log(x)[/tex] simply.
For this case, we have to simplify the expression [tex]log_{0.5}(16)[/tex]
Let this is equal to 'a', then:
[tex]a = \log_{0.5}(16)\\\\0.5^a = 16\\(\dfrac{1}{2})^a = 16\\\\1^a/2^a = 2^4\\\\1 = 2^4 \times 2^a\\2^0 = 2^{4+a}\\\\0 = 4 + a\\a = -4[/tex]
(we used the property that: [tex]b^0 = 1[/tex] and [tex]k^m \times k^n = k^{m+n}[/tex] )
Thus, the value of the logarithm expression [tex]log_{0.5}(16)[/tex] evaluated is given by : Option A: -4.00
Learn more about logarithm here:
https://brainly.com/question/20835449
