I still don't get how do the foloowing questions:
solve for x: 3^x=15, the answer is log15/log3, how do you get this?
if Logbase4(3) =x, find an expression for logbase2(24), the answer is 2x+3 but I dont get it..
solve for x: 3^x=15, the answer is log15/log3, how do you get this?
if Logbase4(3) =x, find an expression for logbase2(24), the answer is 2x+3 but I dont get it..
-
What a logarithm is just a way of undoing an exponent; it is what is called an inverse function. So a good way of thinking about it is that a while exponential functions are raising stuff to a power, logarithms are the exponent itself. For example:
logbase3(9)=x--->what this in essence stating is 3 to the ? power equals 3--->so the answer would be 2 because 3 to the second is 9.
Now here's is where the rules for logarithms come from. Let's use the same problem that we just did;
logbase3(9)=x---->well 9 is also 3^2, so we can rewrite it like this
logbase3(3^2)=x--->this would make it very obvious because if you are saying "3 raised to the x power equals 3^2, it would clearly be 2, but a very important rule with logarithms is what I will do next. You could also rewrite it like this:
2logbase3(3)=x------>when you have something inside the log raised to a power, you can take that term and put it out in front. Where this comes from is from exponent rules; if you think of logarithms as exponents, what do you do when you take an exponent to it a power like 5^2^2? You multiply the exponents, so 5^4. That is all that is being done when you pull that exponent down in front---> this will still work because logbase3(3) will just be 1 and 2*1=2=x
Now to apply these rules to your problems:
3^x=15---->now remember with algebra what you do to one side you have to do to the other, so we'll take the log of both sides; now log with no base is presumed base 10 because it is used very commonly along with ln which is log base e because both of these numbers are used A LOT in science. The important part is that you have to take the same base log on both sides because if you think about it, it would make sense to take logbase1000000000000 on one side but take logbase7 on the other because it would throw everything out of whack.
logbase3(9)=x--->what this in essence stating is 3 to the ? power equals 3--->so the answer would be 2 because 3 to the second is 9.
Now here's is where the rules for logarithms come from. Let's use the same problem that we just did;
logbase3(9)=x---->well 9 is also 3^2, so we can rewrite it like this
logbase3(3^2)=x--->this would make it very obvious because if you are saying "3 raised to the x power equals 3^2, it would clearly be 2, but a very important rule with logarithms is what I will do next. You could also rewrite it like this:
2logbase3(3)=x------>when you have something inside the log raised to a power, you can take that term and put it out in front. Where this comes from is from exponent rules; if you think of logarithms as exponents, what do you do when you take an exponent to it a power like 5^2^2? You multiply the exponents, so 5^4. That is all that is being done when you pull that exponent down in front---> this will still work because logbase3(3) will just be 1 and 2*1=2=x
Now to apply these rules to your problems:
3^x=15---->now remember with algebra what you do to one side you have to do to the other, so we'll take the log of both sides; now log with no base is presumed base 10 because it is used very commonly along with ln which is log base e because both of these numbers are used A LOT in science. The important part is that you have to take the same base log on both sides because if you think about it, it would make sense to take logbase1000000000000 on one side but take logbase7 on the other because it would throw everything out of whack.
keywords: questions,and,exponents,Logarithm,Logarithm and exponents questions...