use the properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms.
show your work
ln x^2/y^3
Assuming x,y AND z are positive use properties of logariths to write the expression as a single logarithm
logx+log y
1/3logx
4 log(xy)-3 log (yz)
Use change of base to evaluate the logarithm
log tiny 8 175
log small 3 X
To rewrite the expression ln(x^2/y^3) as a sum or difference of logarithms, we can use the properties of logarithms.
Property 1: ln(a) - ln(b) = ln(a/b)
Property 2: ln(a^b) = b * ln(a)
Using Property 1, we can rewrite ln(x^2/y^3) as ln(x^2) - ln(y^3).
Using Property 2, we can further rewrite ln(x^2) - ln(y^3) as 2 * ln(x) - 3 * ln(y).
Therefore, the expression ln(x^2/y^3) can be written as a sum or difference of logarithms as 2 * ln(x) - 3 * ln(y).
To rewrite logx + logy as a single logarithm, we can use Property 1 mentioned above.
Property 1: log(a) + log(b) = log(a * b)
Using Property 1, we can rewrite logx + logy as log(x * y).
Therefore, logx + logy can be written as a single logarithm as log(xy).
To rewrite 1/3logx as a single logarithm, we can use Property 2 mentioned above.
Property 2: b * log(a) = log(a^b)
Using Property 2, we can rewrite 1/3logx as log(x^(1/3)).
Therefore, 1/3logx can be written as a single logarithm as log(x^(1/3)).
To rewrite 4 log(xy) - 3 log(yz) as a sum or difference of logarithms or multiples of logarithms, we can use Property 1 mentioned above.
Property 1: a * log(b) = log(b^a)
Using Property 1, we can rewrite 4 log(xy) as log((xy)^4) and 3 log(yz) as log((yz)^3).
Therefore, 4 log(xy) - 3 log(yz) can be written as a sum or difference of logarithms or multiples of logarithms as log((xy)^4) - log((yz)^3).
To evaluate the logarithm using the change of base formula, we can use the following formula:
log base a (b) = log base c (b) / log base c (a)
Using the change of base formula, we can evaluate log tiny 8 175 as log base 10 (175) / log base 10 (8).
Similarly, we can evaluate log small 3 X as log base 10 (X) / log base 10 (3).
By substituting the respective values, we can calculate the values of the logarithms.
To write the expression ln(x^2/y^3) as a sum or difference of logarithms or multiples of logarithms, we can use the properties of logarithms.
1. ln(x^2/y^3) = ln(x^2) - ln(y^3)
Applying the division property of logarithms, we can split the division into subtraction.
Answer: ln(x^2) - ln(y^3)
2. Assuming x, y, and z are positive, we can simplify the expression log(x) + log(y).
Applying the multiplication property of logarithms, we can combine the logarithms.
Answer: log(xy)
3. To simplify the expression (1/3)log(x), we can use the power property of logarithms.
According to the power property, the coefficient in front of the logarithm becomes the exponent of the argument.
Answer: log(x^(1/3))
4. For the expression 4log(xy) - 3log(yz), we can use the power and multiplication properties of logarithms.
According to the multiplication property, we can split the multiplication inside each logarithm.
According to the power property, the coefficient in front of the logarithm becomes the exponent of the argument.
Answer: log((xy)^4) - log((yz)^3)
To evaluate the given logarithms using the change of base formula:
5. To evaluate log₈(175), we can change the base to a common logarithm (base 10) using the change of base formula.
The change of base formula states that logₐ(b) = log(b) / log(a).
Answer: log(175) / log(8)
6. To evaluate log₃(x), we can change the base to a natural logarithm (base e) using the change of base formula.
The change of base formula states that logₐ(b) = log(b) / log(a).
Answer: log(x) / log(3)
there are 3 log properties you will need for the first four.
1. LogA + logB = log(AB)
2. logA - logB = log(A/B)
3. nlogA = log(A^n)
for log8175
= log 175/ log 8 , remember when no base is written, assume base 10
= 2.4837 appr.