how do you express in a single log and simplify
3/2 ln 4x^10 - 4/5 ln 2y^25
(3/2)ln 4 x^10 = (3/2)ln 4 + (30/2) ln x
but ln 4 = ln 2^2
so
3 ln 2 + 15 ln x
now the second term
- 4/5 ln 2y^25
-(4/5) ln 2 - 4 ln y
combining
(11/5) ln 2 + 15 ln x - 4 ln y
= ln [ 2*(11/5) x^15 /y^4 ]
check my arithmetic !
looks good, though I'd have written it
22x^15 / 5y^4
To express and simplify the expression 3/2 ln(4x^10) - 4/5 ln(2y^25) into a single logarithm, we can use the properties of logarithms.
1. Start with the given expression:
3/2 ln(4x^10) - 4/5 ln(2y^25)
2. Apply the power rule of logarithms, which states that the logarithm of a number raised to a power is equal to the product of that power and the logarithm of the number:
3/2 (ln(4) + 10ln(x)) - 4/5 (ln(2) + 25ln(y))
3. Distribute the coefficients:
3/2 ln(4) + 15 ln(x) - 4/5 ln(2) - 20 ln(y)
Note: I simplified the coefficients (3/2 * 10 = 15 and 4/5 * 25 = 20) and used the fact that ln(x^a) = a ln(x) for each term.
Now, we have the expression in a single logarithm, which is:
ln(4^(3/2) * x^15 / 2^(4/5) * y^20)
To further simplify, you can evaluate the logarithmic expression if you have specific values for x and y.