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# Logarithmic equations: variable in the base

## Video transcript

solve for is equal to log base B of 81 for B so let's just remind ourselves what this equation is saying this is saying if I if I raise B to the fourth power then I'm going to get 81 let me rewrite that so if I raise B that's our base so if I raise our base B I guess that's why they picked B B for base if I raise B to the fourth power I'll do the fourth power in orange if I raise B to the fourth power that is going to be equal to 81 I've just rewritten this equation this logarithmic equation I've rewritten it as an exponential equation so it says B to the fourth power is equal to 81 and we still need to solve we need to still solve for B so you just have to think what number you have to multiply by itself four times in order to get 81 and 81 you might jump out of you it is a perfect square we know that 9 times 9 is equal to 81 or we know that another way to say that it is 9 squared is equal to 81 but we have to raise something to the 4th power but 9 itself is 3 times 3 9 itself is 3 times 3 so 1 9 is 3 times 3 and then you multiply it by another 9 that's another 3 times 3 that will also be equal to 81 and we can verify it 3 times 3 is 9 9 times 3 is 27 27 times 3 is 81 so this is 3 to the 4th power 9 squared is the same thing is the same thing as 3 to the 4th power so there we've done it we've said well some number to the 4th power is equal to 81 we know that 3 to the 4th power is equal to 81 so we know that B that B is equal to B is equal to 3 3 to the 4th power is 81 or we could say log base 3 of 81 this is saying what power do I have to raise 3 to to get 81 well we know you have to raise 3 to the fourth power to get to 81 and if you know about fractional exponents and don't worry about this if this confuses you a little bit you could raise both of these if you know about fractional exponents and exponent properties you could do it this way as well you could take C a B to the fourth power is equal to 81 you could raise both of these to the 1/4 power anything you do to one side of equation you have to do the other side and from our exponent properties you know that if you raise something to a power then raise that to a power that's like raising it to the 4 times 1/4 power or this is essentially just raising it to the first power so on the left hand side you're just left with B and on the right hand side your left is equal to 81 to the 1/4 power but figuring out what 81 to the 1/4 power you really have to go through this exercise anyway because when you raise something to the 1/4 power you're really saying well what what do I have to multiply what do I have to raise to the fourth power to get to 81 and then you get what 81 is to the 1/4 power so actually this is another way of realizing that 81 to the 1/4 power is equal to 3 3 to the fourth power is equal to 81 81 to the 1/4 power is equal to 3 but this is the this is you know if this is confusing to you don't worry about it for now the important thing is just you understand what the logarithm is actually saying what after 8 if I raise B to the fourth power I get 81