Assume that x, y, z and b are positive numbers. Use the properties of logarithms to write the expression logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z.
logb(4) sqrt((x^7 y^2)/(z^4))
well, I'm not sure what logb(4) has to do with it, but
log(a^n) = n*log(a)
so,
log(√(x^7y^2/z^4)) = 1/2 (7logx + 2logy - 4logz)
= 7/2 logx + logy - 2logz
regardless of the base.
Well, well, well, let's have some logarithmic fun, shall we? Okay, so we want to express logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z.
First, let's break it down step by step. The expression involves a square root, so remember that we can bring the exponent down as a fraction. This gives us:
logb(4) ((x^7 y^2)/(z^4))^(1/2)
Next, let's bring that exponent down in front of the logarithm, which gives us:
(1/2) logb(4) ((x^7 y^2)/(z^4))
Now, let's tackle the numerator and denominator separately. Using the properties of logarithms, we can write x^7 as logb(x)^(7), y^2 as logb(y)^(2), and z^4 as logb(z)^(4). This gives us:
(1/2) [logb(x^7) + logb(y^2) - logb(z^4)]
But wait, there's more! Using another property of logarithms, we can bring those exponents outside of their respective logarithms. So let's simplify it even further:
(1/2) [7logb(x) + 2logb(y) - 4logb(z)]
And voila! Finally, we have expressed logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z as:
(1/2) [7logb(x) + 2logb(y) - 4logb(z)]
Ta-da! Remember, logarithms can be a bit tricky, but with a touch of clownish humor, we can make it a fun ride!
To write the expression logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z, we can use the properties of logarithms.
First, let's simplify the expression inside the square root:
sqrt((x^7 y^2)/(z^4))
Now, let's write this expression in terms of logarithms:
logb(sqrt((x^7 y^2)/(z^4)))
Next, we can use the property that the square root of a number can be written as raising the number to the power of 1/2:
logb(((x^7 y^2)/(z^4))^(1/2))
Applying the property of logarithms where the logarithm of a division is equal to the logarithm of the numerator minus the logarithm of the denominator:
logb(x^7 y^2) - logb(z^4)
Using the property that the logarithm of a number raised to a power is equal to the power times the logarithm of the number:
7logb(x) + 2logb(y) - 4logb(z)
Therefore, the expression logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z is:
7logb(x) + 2logb(y) - 4logb(z)
To write the expression logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z, we can use the logarithmic properties to simplify it step by step.
Step 1: Simplify the numerator and denominator separately.
In the numerator, we have (x^7 y^2). We can rewrite this as x^7 * y^2.
Similarly, in the denominator, we have z^4.
Step 2: Apply the properties of logarithms to the simplified expression.
According to the log rules, the square root can be expressed as a fractional exponent:
sqrt((x^7 y^2)/(z^4)) = ((x^7 y^2)/(z^4))^(1/2)
Step 3: Move the exponent in front to separate it into individual logarithms.
Using another logarithmic property, we can bring the exponent in front as a coefficient of the logarithm:
logb(((x^7 y^2)/(z^4))^(1/2)) = (1/2) * logb((x^7 y^2)/(z^4))
Step 4: Use the logarithmic property that allows breaking down a quotient into differences to further simplify.
logb((x^7 y^2)/(z^4)) = logb(x^7 y^2) - logb(z^4)
Step 5: Apply the logarithmic properties to separate out the factors.
Using another logarithmic rule, the power of a product can be written as a sum of logarithms:
logb(x^7 y^2) - logb(z^4) = 7 * logb(x) + 2 * logb(y) - 4 * logb(z)
So, the expression logb(4) sqrt((x^7 y^2)/(z^4)) can be written in terms of the logarithms x, y, and z as:
(1/2) * (7 * logb(x) + 2 * logb(y) - 4 * logb(z))