My friend swears by your site so I thought I'd confirm some of my more difficult answers for my math practice exam. Is the solution set for 8^x = 16^ x+2 is it -2? Next is the solution set for log subscript 5 (2x + 3) = 2 is the answer 7/2? Lastly is the answer to Given: log subscript a 2 = .4, log subscript a 3 = .5, log subscript a 5 = .8 what is the value of log subscript a 60 is the value 1.46? Thanks!
Is the solution set for 8^x = 16^ x+2 is it -2? No.
Next is the solution set for log subscript 5 (2x + 3) = 2
is the answer 7/2
No.
Lastly is the answer to Given: log subscript a 2 = .4, log subscript a 3 = .5, log subscript a 5 = .8 what is the value of log subscript a 60 is the value 1.46? Thanks!
60 = 5*2*3*2
loga 60 = loga 5 +loga 3 + 2loga 2 so
No again.
To solve for the solution set of 8^x = 16^(x+2), we can use the property of exponents that states a^b = c^b if and only if a = c.
Let's start by simplifying both sides of the equation:
8^x = (2^3)^x = 2^(3x)
16^(x+2) = (2^4)^(x+2) = 2^(4(x+2)) = 2^(4x+8)
Now we have the equation 2^(3x) = 2^(4x+8). Since the bases are the same, the exponents must be equal:
3x = 4x + 8
To solve for x, we subtract 4x from both sides:
- x = 8
Finally, we multiply both sides by -1 to isolate x:
x = -8
Therefore, the solution set for 8^x = 16^(x+2) is x = -8.
Now let's move on to the equation log subscript 5 (2x + 3) = 2.
To solve for x, we can rewrite the equation using the definition of logarithms:
log subscript a b = c is equivalent to a^c = b.
Applying this definition to the given equation, we have:
5^2 = 2x + 3
Simplifying further:
25 = 2x + 3
Next, we subtract 3 from both sides:
22 = 2x
To isolate x, we divide both sides by 2:
x = 11
Therefore, the correct solution for log subscript 5 (2x + 3) = 2 is x = 11.
Lastly, let's determine the value of log subscript a 60 using the given information.
We know that 60 can be expressed as a product of its prime factors: 60 = 5*2*3*2.
Using the properties of logarithms, we can rewrite the equation step by step:
log subscript a 60 = log subscript a (5*2*3*2)
log subscript a 60 = log subscript a 5 + log subscript a 2 + log subscript a 3 + log subscript a 2
Since we are given log subscript a 2 = 0.4, log subscript a 3 = 0.5, and log subscript a 5 = 0.8, we substitute these values into the equation:
log subscript a 60 = 0.8 + 0.4 + 0.5 + 0.4
Simplifying:
log subscript a 60 = 2.1
Therefore, the value of log subscript a 60 is 2.1.
I hope this helps clarify the correct solutions for the given equations. If you have any more questions, feel free to ask!