Use the Laws of logarithms to rewrite the expression ln(x^3 sqrt(y^4/z^10))
in a form with no logarithm of a product, quotient or power.
After rewriting we have
A ln(x)+B ln(x)+C ln(x)
What is A B and C?
We can start to take it apart using the law of logs.
ln x^3 + 1/2 ln(y^4 - z^10)
3lnx + 1/2 (4)(lny)-1/2 (10) ln z
Can you finish from here.
Remember exponents become multiplications
Multiplications can be changed to additions
and
Divisions can be changed to subtractions.
To rewrite the expression ln(x^3 sqrt(y^4/z^10)) without logarithms of product, quotient, or power, we can use the following log laws:
1) ln(xy) = ln(x) + ln(y) (logarithm of a product)
2) ln(x/y) = ln(x) - ln(y) (logarithm of a quotient)
3) ln(x^n) = n * ln(x) (logarithm of a power)
Let's break down the expression step-by-step:
ln(x^3 sqrt(y^4/z^10))
First, let's rewrite the square root as a fractional exponent:
ln(x^3 (y^4/z^10)^(1/2))
Now, apply the power property of logarithms:
ln(x^3) + ln((y^4/z^10)^(1/2))
Next, simplify the expression within the logarithm:
ln(x^3) + ln(y^2/z^5)
Now apply the power property again to the first term:
3 ln(x) + ln(y^2/z^5)
Now we have the expression in the form A ln(x) + B ln(y) - C ln(z), with A = 3, B = 0 (since there is no ln(y) term), and C = 5 (from the exponent of z).
So, the expression ln(x^3 sqrt(y^4/z^10)) can be rewritten as:
3 ln(x) + 0 ln(y) - 5 ln(z)
To rewrite the expression ln(x^3 sqrt(y^4/z^10)) in a form with no logarithm of a product, quotient, or power, we can make use of the properties of logarithms.
Here are the properties we need to apply:
1. Logarithm of a product: log(ab) = log(a) + log(b)
2. Logarithm of a quotient: log(a/b) = log(a) - log(b)
3. Logarithm of a power: log(a^b) = b * log(a)
Now, let's simplify the given expression step by step:
ln(x^3 sqrt(y^4/z^10))
Step 1: Apply the property of a power on x^3:
= ln((x^3)(sqrt(y^4/z^10)))
Step 2: Apply the property of a square root to the term sqrt(y^4/z^10):
= ln(x^3) + ln(sqrt(y^4/z^10))
Step 3: Apply the property of a power on x^3:
= 3ln(x) + ln(sqrt(y^4/z^10))
Step 4: Apply the property of a square root to the term sqrt(y^4/z^10):
= 3ln(x) + ln(y^4/z^10)^(1/2)
Step 5: Apply the property of a power on (y^4/z^10)^(1/2):
= 3ln(x) + (1/2)ln(y^4/z^10)
Finally, we have the expression rewritten in the desired form:
ln(x^3 sqrt(y^4/z^10)) = 3ln(x) + (1/2)ln(y^4/z^10)
So, A = 3, B = 1/2, and C = 0 since there is no Ln(z) term.