Is log_3 (5) equal to log_5 (3)? Explain your answer. Do not evaluate the logarithms.
To determine if log base 3 of 5 is equal to log base 5 of 3, we can use the property of logarithms known as the change of base formula. The change of base formula allows us to express any logarithm in terms of logarithms with a different base.
The change of base formula states that for any positive number b (≠ 1), and any positive numbers x and y, log base x of y can be expressed as log base b of y divided by log base b of x.
In this case, let's use the change of base formula to express log base 3 of 5 in terms of log base 5:
log base 3 of 5 = (log base 5 of 5) / (log base 5 of 3)
Since the log base 5 of 5 is equal to 1 (since any number raised to the power of one is itself), the expression simplifies to:
log base 3 of 5 = 1 / (log base 5 of 3)
As a result, we can conclude that log base 3 of 5 is not equal to log base 5 of 3, as their expressions involve different bases.