Use the Laws of Logarithms to rewrite the expression
log⌄3(x^3 sqrt3 y^2)
in a form with no logarithm of a product, quotient or power. After rewriting we have
log⌄3(x^3 sqrt3 y^2)=A log⌄3 x + B log⌄3 y
with the constant
A=
and the constant
B=
assuming logs base 3, we have
log x^3 + 1/2 log3 + 2logy
= 3logx + 2logy + 1/2
What do you want to do with the extra 1/2 ?
To rewrite the expression using the laws of logarithms, we can apply the following rules:
1. logₐ(b ⋅ c) = logₐ(b) + logₐ(c)
2. logₐ(b / c) = logₐ(b) - logₐ(c)
3. logₐ(bᶜ) = c ⋅ logₐ(b)
Given the expression log₃(x³ √₃ y²), we can use the laws of logarithms to rewrite it:
log₃(x³ √₃ y²)
= log₃(x³ ⋅ y √₃)
= log₃(x³) + log₃(y √₃)
= 3 log₃(x) + log₃(y √₃)
We can simplify further by applying rule 3 to the second term:
= 3 log₃(x) + log₃(y) + log₃(√₃)
= 3 log₃(x) + log₃(y) + 0.5 log₃(3)
As log₃(√₃) is equal to 0.5 according to the logarithmic identity, we can simplify it to:
= 3 log₃(x) + log₃(y) + 0.5
Therefore, A = 3 and B = 1.
To rewrite the expression log⌄3(x^3 sqrt3 y^2) using the laws of logarithms, we can apply the following rules:
1. Logarithm of a product: log⌄a(xy) = log⌄a(x) + log⌄a(y)
2. Logarithm of a power: log⌄a(x^b) = b * log⌄a(x)
3. Logarithm of a square root: log⌄a(sqrt(b)) = (1/2) * log⌄a(b)
Using these rules, we can break down the given expression step by step:
log⌄3(x^3 sqrt3 y^2)
= log⌄3(x^3) + log⌄3(sqrt3) + log⌄3(y^2)
Now let's simplify each term:
1. log⌄3(x^3): Using the power rule, we have 3 * log⌄3(x)
2. log⌄3(sqrt3): Using the square root rule, we have (1/2) * log⌄3(3)
Note that log⌄3(3) is equal to 1, since 3 raised to what power gives 3? The answer is 1.
Therefore, this term simplifies to (1/2) * 1, which is just 1/2.
3. log⌄3(y^2): Using the power rule, we have 2 * log⌄3(y)
Combining these simplified terms, we get:
log⌄3(x^3 sqrt3 y^2) = 3 * log⌄3(x) + 1/2 + 2 * log⌄3(y)
Therefore, we have A = 3 and B = 2.
So the final rewritten form is:
log⌄3(x^3 sqrt3 y^2) = 3 * log⌄3(x) + (1/2) + 2 * log⌄3(y)