Video transcript
We ‘re asked to solve the log of x plus log of 3 is adequate to 2 log of 4 minus log of 2. thus let me just rewrite it. So we have the log of ten plus the log of 3 is peer to 2 times the logarithm of 4 minus the log of 2, or the logarithm of 2. And this is a reminder. Whenever you see a logarithm written without a establish, the implicit base is 10. So we could write 10 here, 10 here, 10 here, and 10 here. But for the perch of this exemplar, I ‘ll just skip writing the 10 just because it ‘ll save a little bite of time. But remember, this literally means log foundation 10. so this formula, right over here, is the office I have to raise 10 to to get x, the exponent I have to raise 10 to to get 3. nowadays with that out of the way, let ‘s see what logarithm properties we can use. So we know, if we — and these are all the same base — we know that if we have log base a of b plus log base a of degree centigrade, then this is the same thing as log al-qaeda a of bc. And we besides know — so let me write all the logarithm properties that we know over here. We besides know that if we have a logarithm — let me write it this way, actually — if I have b times the log base a of speed of light, this is peer to log base a of c to the bth exponent. And we besides know, and this is derived actually straight from both of these, is that if I have log floor a of boron minus log base a of speed of light, that this is equal to the log base a of bacillus over c. And this is very straight derived from these two correct over here. nowadays with that out of the way, let ‘s see what we can apply. sol right over hera, we have all the logs are the same infrastructure. And we have logarithm of ten plus logarithm of 3. indeed by this property justly over here, the sum of logarithm with the same base, this is going to be peer to log base 3 — blue, logarithm base 10 — then I’ll barely write it here. log base 10 of 3 times x, of 3x. then, based on this property justly over here, this thing could be rewritten — so this is going to be peer to — this thing can be written as logarithm base 10 of 4 to the second world power, which is in truth merely 16. so this is precisely going to be 16. And then we still have minus logarithm root 10 of 2. And now, using this last place, we know we have one logarithm minus another logarithm. This is going to be equal to log foundation 10 of 16 over 2, 16 divided by 2, which is the same thing as 8. So the right-hand side simplifies to log basal 10 of 8. The left side is log base 10 of 3x. thus if 10 to some exponent is going to be equal to 3x. And 10 to the lapp might is going to be adequate to 8. so 3x must be equal to 8. 3x is equal to 8, and then we can divide both sides by 3. Divide both sides by 3, you get ten is equal to 8 over 3. One means, this little footstep here, I said, look, 10 to the — this is an exponent. If I raise 10 to this exponent, I get 3x, 10 to this exponent, I get 8. thus 8 and 3x must be the lapp thing. One other room you could have thought about this is, let ‘s take 10 to this power, on both sides. So you could say 10 to this power, and then 10 to this power over here. If I raise 10 to the exponent that I need to raise 10 to to get to 3x, well, I ‘m precisely going to get 3x. If I raise 10 to the exponent that I need to raise 10 to to get 8, I ‘m just going to get 8. indeed once again, you’ve got the 3x is equal to 8, and then you can simplify. You get adam is equal to 8/3.
