The problem I have to solve is log with base 2 ^6 multiply by log base 6 ^ 8.
I use the change of base formula and got log6/log2 * log8/log6
Which become
log6/log2 * log2()^3/ log 6
I'm stuck here thanks.
you are correct in that you wind up with
log6/log2 * log8/log6
Now, you almost had it, since you recognized (I think) that 8 = 2^3. So, you have
= log8/log2
= 3log2/log2
= 3
To simplify the expression log6/log2 * log2^3/log6, we can use the properties of logarithms.
First, let's simplify log6/log2. By using the change of base formula, we can rewrite it as log(base 2)6:
log6/log2 = log(base 2)6
Next, let's simplify log2^3/log6. By using the change of base formula again, we can rewrite it as log(base 6)2^3:
log2^3/log6 = log(base 6)2^3
Now, we can simplify the expression further:
log(base 2)6 * log(base 6)2^3
Since the base of the first logarithm is 2 and the base of the second logarithm is 6, we cannot simply combine them. However, we can use the property of logarithms: log(base a)b = log(base c)b / log(base c)a.
Applying this property, we get:
log(base 2)6 * (log(base 2)2^3 / log(base 2)6)
The logarithm of any number to the base of the number itself is always 1. Therefore, log(base 2)2 = 1. We can substitute this value to simplify further:
log(base 2)6 * (1^3 / log(base 2)6)
Now, we have:
log(base 2)6 * (1 / log(base 2)6)
The logarithm of a number to the base of the number itself is always 1. Therefore, log(base a)a = 1. We can use this property to simplify further:
log(base 2)6 * (1 / 1)
Finally, we have:
log(base 2)6 * 1
The value of log(base 2)6 depends on the specific notation used. If you have a calculator or computer application that supports logarithmic calculations, you can enter log(base 2)6 to find its numerical value.