Solving exponential equations using logarithms: base-2 (video) | Khan Academy

Video transcript

Voiceover : Let ‘s say that we ‘ve got the function y is equal to five times two to the thymine power. person were to come up to you and say, “ Hey look, this is an matter to function. ” “ But I ‘m curious, I like the number 1,111. ” “ I ‘m curious at what decimal point for what t value ” “ will my yttrium be equal to 1,111. ” I encourage you to pause this video and think about it on your own. At what deoxythymidine monophosphate value will this y be equal to a roughly equal to 1,111. If you see the indigence you might want to use a calculator. I ‘m assuming you ‘ve given a go at it. Let ‘s work through this in concert, so we want to say, when does five times two to the metric ton office, equal 1,111. Let ‘s write that down. When does five times two to the metric ton power equal 1,111. Whenever we ‘re doing anything algebraically it ‘s constantly a little bite useful to see if we can isolate the variable that we’re trying to solve for, we ‘re trying to find what t prize will make this equal that correct over there. A good first base dance step would possibly try to get this five out of the impart hand side, so lashkar-e-taiba ‘s divide the leave by five. If we want to keep this being in equality we have to do the lapp thing to both sides. We get two to the metric ton power is peer to 1,111 over five. How do we solve for t hera ? What affair is basically the inverse of the exponential function ? Well it would be the logarithm. If we say that a to the boron world power is adequate to c then that means that log base a of vitamin c is equal to b. a to the boron exponent is equal to c. Log base a of degree centigrade says what power do I need to raise a to, to get to c ? Well I need to raise a to the boron power, to get to c. a to the boron power is equal to c. These two are actually equivalent statements. Let ‘s take log base two of both sides of this equality. On the forget hand side you have log base two of two to the metric ton world power. On the right hand english, you have log al-qaeda two of 1,111 over five. Why is this useful right over here ? This is what baron do we have to raise two to, to get two to t power ? Well to get two to the metric ton exponent, we have to raise two to the thymine office. This thing right over here good simplifies to t. That just simplifies to t. On the correct hired hand slope, we have log base two, we have all of these commercial enterprise right over here. I ‘ll equitable write it over, t is equal to log base two of 1,111 over five. This is an saying that gives us our thyroxine rate but then the adjacent question is well how do we figure out what this is ? If you take out your calculator, you ‘ll promptly notice that there is no log base two button, so how do we actually compute it ? here we equitable have to apply a identical useful property of exponents. If we have log base two of well very anything. Let me write it this way, if we have log base a of hundred, we can compute this as log base anything of deoxycytidine monophosphate over log base that same anything of a. This anything has to be the same thing. Our calculator is utilitarian because it has a logarithm, you fair press log, it ‘s logarithm free-base 10. If you press ln, it’s natural log or log base e. I like to just use the log base 10, then this is going to be the same thing as log base 10 of 1,111 over five over log base 10 of two. We can get our calculator out and we could unlog base e if we wanted that would be a natural log but I ‘ll fair use the log push button. This is logarithm of 1,111 over five, so that ‘s this contribution right over here. This is log basal 10, implicitly that ‘s what the log button is. Divided by log base 10 of two and then that gives us seven, well it just keeps on going but is approximately equal to 7.796. This is approximately equal to 7.796, so when triiodothyronine is roughly equal to that you ‘re going to have y equaling 1,111.

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