in this lesson we're going to focus on solving basic logarithmic equations let's start with this one log base 2 of 16 is equal to x go ahead and find the value of x what we need to do is convert it to exponential form 2 raised to the x is equal to 16. now you might already see the answer but i'm going to go ahead and solve it now we know that 2 to the 4th power is 16. so therefore x is equal to four you can also do this log of 16 divided by log of two if you type that in that's equal to four which is x using the change of base formula try this one log of x log base x of 81 is equal to four what is x so let's convert it to its exponential form x to the fourth is 81. so now let's find the value of x to do that we need to take the fourth root of both sides so the fourth root of 81 is equal to 3 because three to the fourth is eighty-one now what about this one log base five of x is equal to three convert it to its exponential form five to the third is equal to x that's five times five times five five times five is twenty-five and five quarters is a dollar twenty-five so twenty-five times 5 is 125. let's try some more examples log base 32 of x let's say it's equal to 4 over 5. go ahead and try this one so 32 raised to the 4 over 5 is equal to x and what is that equal to what is 32 raised to the 4 over 5 how can we simplify this exponential fraction so first you want to find the fifth root of 32 and then raise it to the fourth power so what number multiplied it by itself five times is 32 2 to the fifth power is 32 so therefore 32 to the one fifth is two so now we got to find out what two to the fourth is equal to two to the fourth is sixteen and so x is sixteen here's the next one log base three of five x plus one is equal to four find the value of x so let's convert it to its exponential form three to the fourth is equal to what's inside and three to the fourth is eighty-one let's subtract both sides by one eighty-one minus one is eighty and if we divide by five 80 divided by 5 is 16. so that's the value of x if log x is equal to 24 what is the value of x now we don't have a base if there are no bases written it's assumed to be base 10. so this is going to be 10 raised to the 24th which is a very very big number so i'm going to leave it as 10 to the 24th now what about this one ln x is equal to 7 what is the value of x natural logs have the base e so e to the 7th is equal to x and that is the answer now if you want the decimal value x is approximately 1096.65 try this one log base 7 x squared plus 3x plus 9 is equal to 2. so let's convert it to its exponential form 7 raised to the second power is equal to what's inside that is it's equal to x squared plus three x plus nine seven squared is forty nine and let's subtract both sides by nine so forty is equal to x squared plus three x let's move the 40 to the right side so x squared plus 3x minus 40 is equal to zero now what two numbers multiply to negative 40 but add to three this is going to be 8 and negative 5. so this can be factored to x plus 8 times x minus 5. so x is equal to negative 8 and positive 5. let's try this example the natural log of three x minus two let's say it's equal to five what is the value of x so keep in mind the base is e so e raised to the fifth power is equal to the stuff inside that is it's equal to three x minus two so now let's get x by itself so let's add two to both sides so e to the fifth plus two is equal to three x and if we divide by three x is equal to e to the fifth plus two over three and as a decimal this is approximately 50.14 let's try one more example four times the natural log of two x minus one plus three is equal to eleven go ahead and find the value of x so let's begin by subtracting both sides by three so eleven minus three is eight now let's divide both sides by four eight divided by four is two at this point we can convert it into its exponential form so e raised to the second power is equal to 2x minus 1. so let's add one to both sides and then let's divide by two so x is equal to e squared plus one over two and as a decimal that is approximately 4.1945 consider this equation log base 3 5 x plus 2 and let's say that's equal to log base 3 7 x minus eight what is the value of x now because the bases are the same that means that the stuff inside of the log must be equal to each other so therefore five x plus two is equal to seven x minus eight so now we can find the value of x let's subtract both sides by five x and let's add eight to both sides two plus eight is ten seven x minus five x is two x so therefore x is ten divided by two which is equal to five and that's all you need to do for this example so now it's your turn try this example let's say that log base 2 of x squared plus 4x is equal to log base 2 of 5. feel free to pause the video so because the bases are the same that means that x squared plus four x must equal five now we have a quadratic equation so let's move the five to the left side it's going to be negative five on the left and now let's factor what two numbers multiply to negative five but add to the middle term four this is positive five and negative one so we have x plus five times x minus one is equal to zero so x is equal to negative five and x is equal to one and so those are the answers now let's work on a different example log base 2 of x plus log base 2 of x plus 4 is equal to 5. go ahead and calculate the value of x now let's review a basic property of logs log a plus log b is equal to log a times b we need to use this equation to combine the two logs into a single log so if log base 2 of x plus log base 2 of x plus 4 is equal to 5 then log base 2 of x times x plus 4. that has to be equal to 5 as well so now let's convert it to its exponential form 2 to the fifth is equal to what's inside and let's distribute the x so x times x is x squared x times four is four x and two to the fifth power is thirty two so now let's move the 32 to the other side it's positive on the left side which means it's negative on the right side so now let's factor what two numbers multiply to negative 32 but add to four so we have eight and negative four so this is going to be x plus eight times x minus four so x is equal to negative eight and positive four in this example we need to check the answers we need to check for extraneous solutions if we replace x with negative eight notice that this is not going to work you cannot have a negative number inside a log and you could type it in your calculator type in log negative eight it's going to give you an error so this answer is it doesn't work now let's plug in 4. so log base 2 of 4 plus log base 2 of 4 plus 4 which is 8 is that equal to 5. well we know 2 squared is 4. so log base 2 of 4 is 2 log base 2 of 8 is 3 because 2 to the third power is 8 and 2 plus 3 isn't 8 5. so that works try this one log base 3 of x plus 1 let's say it's equal to 3 minus log base 3 of x plus 7. so what do you think we need to do here in this example we cannot set x plus one equal to x plus seven because of the three here however we could take this log and move it to this side and it's going to be similar to the last problem if we do that so it's going to be positive log 3 x plus 7. so now we can combine the two logs into a single log by multiplying the stuff on the inside so we need to multiply x plus one times x plus seven so now let's convert it to its exponential form three to the third power is equal to the stuff inside and let's go ahead and foil x plus one times x plus seven x times x is x squared x times seven that's seven x and then we have one times x which is x and one times seven that's seven three to the third power is twenty seven seven x plus x is eight x and now let's subtract both sides by twenty seven so we have 0 on the left and 7 minus 27 is negative 20. so what two numbers multiply to negative 20 but add to the middle number eight this is going to be ten and negative two so we're gonna have x plus ten and x minus two so x is equal to negative ten and positive two so let's see which one works and which one does not work so the original equation is log base three x plus 1 which is equal to everything on the right side so we can see that negative 10 is not going to work if you plug in negative 10 negative 10 plus one is a negative number and we cannot have a negative number inside the log if we plug in two two plus one is three two plus seven is nine that's okay so two is gonna work that's the answer so here's another equation that you'll see as well so this time instead of having a plus sign in between the two logs we're gonna have a minus sign and if you recall log of a minus log of b is equal to log a divided by b so we need to combine the two logs using division so we're going to write it as a single log and the one that's positive is going to go on top that is 2x plus 6. x minus 1 has a negative sign in front of the log and all of this is equal to 1. so now let's convert it to its exponential form four to the first power is equal to what's inside so that's equal to two x plus six divided by x minus one four is the same as four over one so now let's cross multiply 1 times 2x plus 6 is 2x plus 6 and 4 times x minus 1 is 4x minus 4. now let's subtract both sides by 2x and let's add 4 to both sides 6 plus 4 is 10 4x minus 2x is 2x and 10 divided by 2 is 5. so this is the answer and if you plug in 5 into x minus 1 that will give you a positive number if you plug it into here that will also give you a positive number so we won't get any negative numbers inside a log so x is equal to five this is the last example for this video feel free to try so whenever you have an equation with two logs and a number you want to get the two logs by itself on one side of the equation so let's move this to this side so log base 2 x plus 3 minus log base 2 x minus 3 that's equal to 4. so let's combine this two logs into a single log using division the one that's positive will go on top and now let's convert it to its exponential form so 2 to the fourth power is equal to what's inside and two to the fourth is 16 so now let's go ahead and cross multiply so we have x times i mean 1 times x plus 3 which is x plus 3 and then 16 times x minus 3 that's 16x minus 48. now let's subtract both sides by x and let's add 48 to both sides so these will cancel 48 plus 3 that's 51 and let's see 16x minus x is 15x so if we divide by 15 x is 51 over 15. now both numbers are divisible by 3. 51 divided by 3 is 17 15 divided by 3 is 5. so it's 17 over 5. let's say that log x raised to the log x is equal to 49. what is the value of x what would you do first in this problem what we can do is move this in front so we'll have log x times log x is equal to 49 log x times log x is equivalent to log x squared and now what we can do is take the square root of both sides the square root will cancel with the square so on the left it's just going to be log x on the right the square root of 47 i mean not 47 but 49 is plus or minus 7. so log x is equal to positive 7 and log x is equal to negative seven now keep in mind the base is ten so let's convert it into its exponential form ten raised to the seven is equal to x so x is equal to ten to the seventh which is ten million ten to the negative seven is also equal to x and so this is the answer ten to the negative seven is like point zero zero zero zero zero zero one so it's a positive number and if you plug in a positive number into a log there's nothing wrong with that there's no issue with that so both answers are acceptable you can't plug in a negative number inside the log but these are positive numbers one is just very large the other is simply very small here's the next one log of x squared is equal to log x squared what is the value of x in this problem the first thing i would recommend doing is moving the two so this is uh 2 log x you can't move this 2 because the 2 applies to everything this 2 means that you have log x times log x which is completely different so we can't move that 2 to the front so right now this is what we have now i'm going to take 2 log x and move it to the right side so 0 is equal to log x squared minus two log x now let's factor the gcf is log x log x squared divided by log x is log x negative two log x divided by log x is simply negative two now we can set each factor equal to zero so log x is equal to zero and also this part log x minus two is also equal to zero now let's be careful because this is log x minus two is equal to zero the x is part of the log but the 2 is not now the base of log is 10 10 raised to the 0 is equal to x and anything raised to 0 power is 1 so therefore x is equal to 1. now let's get the other answer so first let's add 2 to both sides so log x is equal to 2 and the base is ten so ten squared is equal to x ten squared is equal to a hundred so x is equal to one hundred now let's make sure that we do indeed have the right answers so let's check each one and let's write the original equation which was log x squared is equal to log x squared so let's start with one log one squared is that equal to log one squared log one is zero and zero squared is also zero so that is a true statement so x is indeed equal to one now let's plug in a hundred log one hundred squared is that equal to log 100 squared well we can move the 2 to the front so this is 2 times log 100 and log 100 keep in mind the base is 10 ten squared is a hundred so log one hundred is two so two times two is is equal to two squared they both equal four so x is indeed equal to one and one hundred try this log of log of x is equal to four go ahead and find the value of x if you don't see a base it's base ten so ten raised to the four is equal to what's inside which is log x now ten to the four is ten thousand it's a one with four zeros now keep in mind the base of this log is also 10 and 10 raised to the 10 000 is equal to the stuff inside which is x so that's the answer it's 10 raised to the 10 000. here's another one like that log base 3 of log base 2x let's say that's equal to 2. so 3 raised to the second power is equal to what's inside which is log base 2 of x and 3 squared is 9. and we know that two raised to the nine is equal to the stuff inside which is x so x is two to the ninth power so what exactly is two to the ninth power two to the ninth is two to the fourth times two to the fourth times two to the first power because four plus four plus one is nine two to the fourth is sixteen sixteen times sixteen is two fifty-six two fifty-six times two is 512 and you can also just type this in your calculator you should get 512 as well you