About this video
The video solves the logarithmic equation 4 log_x(1/16) + 11 = 3 by isolating the logarithm (subtract 11, divide by 4) to get log_x(1/16) = -2. It rewrites this in exponential form as x^(-2) = 1/16, raises both sides to the -1 power to get x^2 = 16, and takes the positive root x = 4 because logarithm bases are positive.
You'll be able to
- isolate a logarithm in an equation using inverse operations.
- rewrite a logarithmic equation in exponential form.
- solve for an unknown base and reject a negative base.
Key words
- logarithmic equation
- An equation where the unknown appears inside a logarithm, here as the base.
- base
- The number in log_b(a) that is raised to a power; it must be positive.
- exponential form
- Writing log_b(a) = c as b^c = a.
- reciprocal
- 1 divided by a number; raising to the -1 power gives the reciprocal, so (1/16)^(-1) = 16.
- principal root
- The positive square root of a number, such as 4 for 16.
Transcript
- [Presenter] So let's see if we can solve this logarithmic equation. I call it that because the x is involved in a logarithm right over here. So pause this video. Have a go at this before we do this together. So the key is, is that we really want to isolate this part that involves the logarithm.
So to do that, let's start by subtracting 11 from both sides. Get rid of this 11 on the left-hand side. And so what that does, is on the left-hand side, we're just left with 4, (laughs) I was gonna say flog, 4 log base x of 1/16 is equal to 3-11 is -8. All right, now if we really want to isolate the logarithm part, we can divide both sides by 4, and we are going to get log base x of 1/16 is equal to -8/4, is -2.
So another way to think about this is we could just rewrite this based on the definition of a logarithm. If log base x of 1/16 is equal to -2, this is the same thing as saying that x to the -2 power is equal to 1/16 This is the exponent right over here. And so you might already have a sense of what this might be, but if you want to get rid of this negative in the exponent, we could take both sides to the -1 power.
Then on the left-hand side, if you raise something to a power and then raise that to another power, you can multiply the two exponents. So this is going to be x squared on the left-hand side. That's why I raised it to a -1, essentially to get rid of this negative on the -2.
So we get x squared. And then what's 1/16 to the -1? Well, if you raise something to the -1, you're really just taking the reciprocal of it. So that's going to be 16/1 or just 16. And so the answer there is, "What squared is going to be equal to 16?" If we're assuming, and it's usually the case that you can only have positive bases for the logarithm function, we would say that x is equal to the principal root of 16 which is equal to 4.
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