1. Given log2(m) = x, log4(n) = y
What is mn?
2. Given log5(m) = x, log√5(n) = y
What is mn?
3. Given log2(m^2) = x, log8(√n) = y
What is mn?
m = 2^x
n = 4^y = 2^(2y)
mn = 2^(x+2y)
do the others in like wise, using the properties of logs and exponents.
Come back with your work if you get stuck.
I'll do #2, you do the rest the same way:
log5 m = x <-----> 5^x = m
log√5 n = y <-----> √5^y = n
mn = 5^x * (√5)^y
= (√5)^(2x) * √5^y
= (√5)^(2x+y)
To find the value of mn in each given equation, we'll need to use logarithmic properties and solve for m and n separately. Let's go through each question step by step:
1. Given log2(m) = x and log4(n) = y, we want to find mn.
To solve this, we can rewrite the equations using exponential form:
m = 2^x and n = 4^y.
Now, we can substitute these expressions for m and n into the expression mn:
mn = (2^x) * (4^y).
Since 4 can be expressed as 2^2, we can substitute that into the equation:
mn = (2^x) * (2^(2y)).
By applying the rule of exponents, we can simplify the equation further:
mn = 2^(x + 2y).
2. Given log5(m) = x and log√5(n) = y, we want to find mn.
Similar to the previous question, we'll start by converting the logarithmic expressions to exponential form:
m = 5^x and n = (√5)^y.
Simplifying the expression for n:
n = (5^(1/2))^y = 5^(1/2 * y) = 5^(y/2).
Now, we can substitute these expressions for m and n into the expression mn:
mn = (5^x) * (5^(y/2)).
Using the rule of exponents, the equation can be simplified:
mn = 5^(x + (y/2)).
3. Given log2(m^2) = x and log8(√n) = y, we want to find mn.
Let's rewrite the logarithmic expressions in exponential form:
m^2 = 2^x and √n = 8^y.
Simplifying the expression for √n:
√n = (8^(1/2))^y = 8^(1/2 * y) = 8^(y/2).
We have m^2 = 2^x, and to find m, we need to take the square root of both sides:
m = sqrt(2^x) = (2^x)^(1/2) = 2^(x/2).
Now, we can substitute these expressions for m and √n into the expression mn:
mn = (2^(x/2)) * (8^(y/2)).
Since 8 can be expressed as 2^3, we can substitute that into the equation:
mn = (2^(x/2)) * (2^(3y/2)).
Using the rule of exponents, we can simplify the equation further:
mn = 2^((x/2) + (3y/2)).
So, to find the value of mn in each equation:
1. mn = 2^(x + 2y).
2. mn = 5^(x + (y/2)).
3. mn = 2^((x/2) + (3y/2)).