Why does 2ln6 - 2ln3 = ln4?
2ln6 = ln 36
2ln3 = ln 9
ln36 - ln 9 = ln(36/9) = ln 4
or
2(ln6-ln3) = 2(ln(6/2) = 2ln2 = ln(2^2) = ln 4
thank you!
To understand why 2ln6 - 2ln3 is equal to ln4, we can use the properties of logarithms.
First, let's simplify the expression:
2ln6 - 2ln3
Using the property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the individual factors, we can rewrite this as:
ln(6^2) - ln(3^2)
Simplifying further, we have:
ln(36) - ln(9)
Now, we use another property of logarithms which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. We can write this as:
ln(36/9)
Simplifying the quotient, we get:
ln(4)
Therefore, 2ln6 - 2ln3 simplifies to ln4.
To understand why 2ln6 - 2ln3 equals ln4, let's break down the steps to solve this equation.
Step 1: Apply the logarithmic property. The property states that when subtracting logarithms with the same base, we can divide the inside of the logarithms. In other words, we can rewrite the equation as:
2ln(6/3) = ln4.
Step 2: Simplify the inside of the logarithms.
6/3 simplifies to 2, so the equation becomes:
2ln2 = ln4.
Step 3: Apply the property of logarithms again. The property states that when a coefficient is multiplied to the logarithm, it can be brought up as an exponent. In this case, the equation becomes:
ln(2^2) = ln4.
Step 4: Simplify the equation.
2^2 is equal to 4, so the equation becomes:
ln4 = ln4.
Since the natural logarithm function (ln) is the inverse of the exponential function, ln4 is equal to ln4. Therefore, 2ln6 - 2ln3 is equal to ln4.