Find the exact value of the expression

Question

Answer 1

Given a log expression to solve, the first step you always want to take is to see if you can spread it out using your log rules. In these next few steps, I will expand the log into its most spread out form and then simplify.

log₄(∛2 / 16) = log₄(∛2) - log₄(16) - Dividing in a log is the same as subtracting two logs with the same base

log₄(∛2) - log₄(16) = (log₄(2) / 3) - log₄(16) - Exponents or roots in a log are the same as multiplying the log by a constant, in this case cube root becomes 1/3.

Now we can simplify each log because we have gotten them to a solvable form. I will only touch on these briefly, but I can explain how a log works if need be.

log₄(2) is equal to 1/2 and log₄(16) is equal to 2. We can plug these back into our simplified formula above:

(log₄(2) / 3) - log₄(16) = ((1/2)/3) - 2 = 1/6 - 2 = -11/6

-11/6 would be the exact value of the expression. Let me know if you need clarification or have any questions!

Answer 2

log4(2^1/3/16) = x

4^x = (2^1/3)/4^2, multiply both sides by 4^2

(4^x)(4^2) = 4^(x+2) = 2^(1/3) = 4^(1/6) = (2^2)^1/3

set the exponents equal. 4^(x+2) = 4^1/6

x+2 = 1/6

x =1/6-2 = 1/6-12/6 = (1-12)/6 = -11/6

x = -11/6

the log to the base 4 of (1/16) the cube root of 2 is -11/6

Answer 3

log₄(2^1/3/16) = log₄(2^1/3) - log₄(4²) = log₄(2)/3 -2log₄(4) = log₄(4^1/2)/3 - 2

= log₄(4)/6 -2= 1/6 - 2 = -11/6

Note: log_x(x) = 1