The question says to make a statement about Log A over B & Log B over A & prove it.
How do I answer?
To answer this question, you will need to make a statement about the logarithms of two numbers: log A over B and log B over A, and then prove it.
Statement: log A over B is equal to the negative value of log B over A.
Proof:
1. Start by writing the definition of logarithms:
log A over B = x ----(1)
log B over A = y ----(2)
2. Rewrite the logarithmic equations using exponential notation:
A = B^x ----(3)
B = A^y ----(4)
3. Substituting equation (3) into equation (4):
B = (B^x)^y
4. Apply the property of exponents, which states that when raising a power to another power, you multiply the exponents:
B = B^(xy)
5. The base B is the same on both sides of the equation, so the exponents must be equal:
1 = xy
6. Rearranging equation (6):
x = 1/y
7. Substituting this value of x into equation (1):
log A over B = 1/y
8. Taking the negative value of 1/y gives us:
-(1/y) = -log B over A
9. Therefore, we can conclude that:
log A over B = -log B over A
This proves the statement that log A over B is equal to the negative value of log B over A.