Determine the value of each of the following logarithmic expressions. Do not change bases.
A) log8 4 <----- 8 is subscript
b)log 1/3 27 <---- 1/3 is subscript
let x = log8 4
8^x = 4
(2^3)^x = 2^2
2^(3x) = 2^2
3x=2
x = 2/3
do the 2nd the same way, let me know what you got
log(27) / log(1/3)
log(3^3) / log(3^-1)
3 log(3)/-1 log(3)
-3
To determine the value of each logarithmic expression, we need to understand how logarithms work and how to evaluate them.
A) log8 4:
To determine the value of log8 4, we need to figure out to what power we need to raise the base (8) in order to obtain the given number (4).
In other words, we need to find the exponent (x) in the equation 8^x = 4.
By using the concept of logarithms, we can rewrite this equation as log8 4 = x. This means that the value of log8 4 is equal to x.
To solve for x, we can rewrite 8^x = 4 in exponential form as 8 = 4^x.
Now, we observe that 8 can be expressed as 2^3. Therefore, we have 2^3 = 4^x.
By equating the exponents, we get 3 = x.
Hence, the value of log8 4 is 3.
B) log(1/3) 27:
To determine the value of log(1/3) 27, we need to find the exponent (x) in the equation (1/3)^x = 27.
Using the concept of logarithms, we can rewrite this equation as log(1/3) 27 = x.
To solve for x, we can rewrite (1/3)^x = 27 in exponential form as (1/3) = 27^x.
Now, we observe that 27 can be expressed as 3^3. Therefore, we have (1/3)^x = (3^3)^x.
By applying the property of exponents (raising a power to a power), we get (1/3)^x = 3^(3x).
Since the bases are the same, the powers must be equal. Thus, we have x = 3x.
This implies 1 = 3x - x = 2x.
By dividing both sides by 2, we find x = 1/2.
Hence, the value of log(1/3) 27 is 1/2.