Logarithmic equations (Algebra 2 level)
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Logarithmic equations: variable in the base
Solve 4 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 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 need to still solve for b. So you just have to think, what number do I have to multiply by itself four times in order to get 81. And 81 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 is, nine squared is equal to 81. But we have to raise something to the fourth power. But 9 itself is 3 times 3. So one 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 fourth power. 9 squared is the same thing as 3 to the fourth power. So there we've done it. We've said, well, some number to the fourth power is equal to 81. We know that 3 to the fourth power is equal to 81. So we know that b is equal to 3. 3 to the fourth power is 81. Or we could say, log base three 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, so you have 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 to 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, you're 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 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 if this is confusing to you, don't worry about it for now. The important thing is you understand what the logarithm is actually saying. If I raise b to the fourth power I get 81.