Algebra

Solving exponential equation | Exponential and logarithmic functions | Algebra II | Khan Academy

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Let’s say we have the function y equal to 5 by (2 to the power of t). And someone comes and tells you: “Look, this seems like an interesting feature, but I’m curious – I like the number 1111, I’m curious at what point, for what t value my y will be equal to 1111. ” I advise you to pause the video and think about it on your own. At what t value will this y be equal to or approximately equal to 1111. If necessary, you can even use a calculator. I accept that he tried. Let’s solve this together now. We wonder when 5 over (2 to the power of t) is equal to 1111. Let’s write this down. When 5 by (2 of degree t) is equal to 1111? When we do something algebraic, it is always useful to see if we can isolate the variable we are trying to find – we are trying to find out what t value this will do equal to this here. A good first step would be to try let’s remove this 5 from the left side, so let’s divide the left side by 5. If I want this to continue to be equality, I have to do the same thing with both sides. We get 2 to the power of t is equal to 1111/5. How to find t? Which function is the opposite of the indicative function? This will be the logarithm. If I say that a of degree b is equal to c, this means that the logarithm of c at base a is equal to b. a of degree b is equal to c. The logarithm of c at the base a tells us to what extent it should to raise a to get c. I have to raise a to power b to get c. a of degree b is equal to c. These two statements are equivalent. Let us make a logarithm at base 2 on both sides of this equation. On the left side you have a logarithm of (2 to the power of t) at base 2. On the right side you have a logarithm of 1111/5 at base 2. Why is this useful here? This is how much we need to raise 2, to get 2 to the power of t. To get 2 to the power of t, we need to raise 2 to the power of t. This thing is here it is simply simplified to t. This is simplified to t. On the right side we have a logarithm at base 2, we have it all here. I’ll just write it down – t is equal to the logarithm of 1111/5 at base 2. This is an expression that gives us the value of t, but the next question is how to calculate how much it is. If you take out your calculator, you will quickly notice that there is no logarithm button at base 2, so how do we calculate it? Here we must apply a very useful property of degrees. If we have a logarithm of whatever at base 2 … Let me write it like this, if we have a logarithm of c at base a, we can calculate this as the logarithm of c on any basis, on a logarithm of a at the base this same thing. As this unknown must be the same. Our calculator is useful because it has a “log” button. you just push it, and that’s the logarithm at base 10. If you press “In”, this is a natural logarithm or a logarithm at the base \’e\’. I prefer to use a logarithm at base 10, so it will be the same thing as a logarithm of 1111/5 at base 10 on a logarithm of 2 at base 10. We can take out our calculator and we can use a logarithm on the base ‘e’ if we want – this will be a natural logarithm, but i will just use the “log” button. This is a logarithm of 1111/5 – this is the part here. This is the logarithm at base 10 – this is indicated by the “log” button. Divided by a logarithm of 2 at base 10 – this gives us 7 with many digits after the decimal point, but is approximately equal to 7,796. This is approximately equal to 7,796 so that when t is approximately equal to that, y will be equal to 1111.

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