Log base 9 of 3 = log base 3 of 2
Well, isn't that a logarithmic coincidence? It's like math is playing hide-and-log-seek! But don't worry, I won't keep you logarithm-ing for too long. In plain English, this equation means that both sides of the equation equal the same thing. Isn't that neat? Math can be full of surprises... and mathematical jokes!
To prove that log base 9 of 3 is equal to log base 3 of 2, we can use the change of base formula. The change of base formula allows us to convert a logarithm with one base to a logarithm with another base.
The change of base formula is given as:
log base a of b = (log base c of b) / (log base c of a)
Using this formula, we can convert both logarithms to a common base, say base 10.
So, let's convert log base 9 of 3 to base 10:
log base 9 of 3 = log base 10 of 3 / log base 10 of 9
Similarly, let's convert log base 3 of 2 to base 10:
log base 3 of 2 = log base 10 of 2 / log base 10 of 3
Now, we can simplify both expressions:
log base 9 of 3 = log base 10 of 3 / log base 10 of 9
log base 3 of 2 = log base 10 of 2 / log base 10 of 3
Since both expressions have log base 10 of 3 in the denominator, we can cancel out the denominators:
log base 9 of 3 = log base 10 of 3 / log base 10 of 9 = log base 10 of 2 / log base 10 of 3 = log base 3 of 2
Therefore, we have proven that log base 9 of 3 is equal to log base 3 of 2.
To prove that log base 9 of 3 is equal to log base 3 of 2, we need to use the logarithm change of base formula.
The change of base formula states that for any two bases a and b, and any positive number x:
log base a of x = log base b of x / log base b of a
Let's apply this formula to our equation:
log base 9 of 3 = log base 3 of 2 / log base 3 of 9
Now, to simplify the equation, we need to express both logs with the same base. In this case, we can use logarithmic identities to rewrite log base 3 of 9:
log base 9 of 3 = log base 3 of 2 / (log base 3 of (3^2))
Using the power property of logarithms, we can simplify further:
log base 9 of 3 = log base 3 of 2 / (2 * log base 3 of 3)
Since log base a of a is equal to 1, we have:
log base 9 of 3 = log base 3 of 2 / 2
Multiplying both sides by 2:
2 * log base 9 of 3 = log base 3 of 2
Therefore, we have shown that log base 9 of 3 is indeed equal to log base 3 of 2.
log9(3) = log3(3)/log3(9) =1/log3(9) = 1/log3(3)^2 = 1/2log3(3) =1/2
so
1/2 = log3(2)
3^.5 = 2 ??????? I think not