Please help I don't understand
Contract the expression this is use properties of logarithm to write each expression as a single logarithm with a coefficient of 1.
In3-3 In 9 +In 18
In3-2 in (9=6)
In 3-2 (In 4 In 8)
Thank you so much
just remember that
lna - lnb = ln(a/b)
lna + lnb = ln(a*b)
ln a^b = b*lna
ln3 - 3ln9 + ln18
ln3 - 3*2ln3 + ln(9*2)
ln3 - 6ln3 + 2ln3 + ln2
-2ln3 + ln2
ln 1/9 + ln2
ln 2/9
#2 has a typo
I suspect a typo in #3 as well, since
ln4 ln8 = ln(8^ln4) or ln(4^ln8)
To contract the given expression using properties of logarithms, we can start by applying the following properties:
1. The product property: log(base a) b + log(base a) c = log(base a) (b * c)
2. The quotient property: log(base a) b - log(base a) c = log(base a) (b / c)
Let's apply these properties step by step to contract the expression:
Step 1: In3 - 3 In 9 + In 18
First, we can use the product property to combine the two logarithms with base 3:
In3 - 3 In 9 = In3 - In(9^3)
Next, applying the quotient property, we can combine the remaining two logarithms:
In3 - In(9^3) + In 18 = In3 - In[(9^3) * 18]
Finally, we can simplify further:
In3 - In[(9^3) * 18] = In3 - In(729 * 18) = In3 - In(13122)
Therefore, the contracted expression is: In3 - In(13122).
Step 2: In3 - 2 In (9=6)
Here, it seems there might be an "=" sign present in the expression, which should not be there for simplification purposes. Assuming it's a typo, let's correct it and simplify the expression.
In3 - 2 In (9 * 6) = In3 - In[(9 * 6)^2]
Simplifying further, we get:
In3 - In[54^2] = In3 - In2916
Therefore, the contracted expression is: In3 - In(2916).
Step 3: In 3 - 2 (In 4 In 8)
In this last expression, we see the coefficient "2" outside the parentheses. To simplify it, we need to apply the product property to the logarithms inside the parentheses:
In 3 - 2 (In 4 * In 8) = In 3 - 2 (In [4 * 8])
Simplifying further:
In 3 - 2 (In [32]) = In 3 - 2 (In 32)
Therefore, the contracted expression is: In 3 - 2 (In 32).
I hope this explanation helps you understand how to contract logarithmic expressions using properties of logarithms!