consider the following exponential equation three raised to the x plus two is equal to nine raised to the two x minus three how can we find the value of x without using logs what you need to do is change base nine into base three three squared is equal to nine so we can replace nine with three squared and whenever you raise one exponent to another exponent you need to multiply so we got to multiply two by two x minus three so on the right side this is going to be three raised to the four x minus six now if the bases are the same then the exponents must be equal to each other therefore x plus 2 is equal to four x minus six now let's subtract both sides by x and let's add six to both sides so these two will cancel two plus six is eight four x minus x is three x so we can see that x is equal to eight over three and so that's the answer for this one now let's work on another example 8 raised to the 4x minus 12 is equal to 16 raised to the 5x minus 3. so go ahead and find the value of x now what we need to do is convert 8 and 16 into a common base 2 is a multiple of 8 and 16 so 2 is the common base 2 to the third power is 8 2 to the fourth power 16. so let's replace 8 with 2 to the third power and let's replace 16 with 2 to the fourth power now we need to multiply 4x minus 12 by 3. 4x times 3 is 12x 3 times negative 12 is negative 36. now we need to multiply 5x minus 3 by 4. so that's going to give us 20x minus 12. so we could set these two equal to each other now that we have the same base so 12x minus 36 is equal to 20x minus 12. let's subtract both sides by 12x let's add 12 to both sides so negative 24 is equal to 8x therefore x is equal to negative three let's try one more example 27 raised to the 3x minus 2. let's say that's equal to 81 raised to the two x plus seven so try this problem now three to the third power is equal to twenty seven and three to the fourth power is equal to eighty one so let's replace twenty seven with three cubed and let's replace eighty-one with three to the fourth power three times three x minus two that's going to be nine x minus six and four times two x plus seven that's eight x plus twenty eight so now that we have the same base we can set the exponents equal to each other so nine x minus six is equal to eight x plus twenty eight so let's subtract both sides by eight x and let's add six to both sides so these will cancel nine x minus eight x is x twenty eight plus six is thirty four so x is equal to thirty four in this example three raised to the x is equal to eight what is the value of x in order to find it we can take the log or the natural log of both sides it really doesn't matter which one you're going to use let's take the log of both sides once you do that you can move the exponent to the front so x log 3 is equal to log 8. so x is log eight divided by log three if you type this in you should get 1.8928 and you can check your answer what is 3 raised to the 1.8928 if you type that in the calculator will give you this answer eight point zero zero zero zero nine granted this is rounded so this is going to be very close to eight what about this one if e raised to the x is equal to 7 what is the value of x now whenever you're dealing with base e it's better to use the natural log as opposed to the regular log so we're going to take the natural log of both sides and we're going to move x to the front so x ln e is equal to ln7 and e is one so x times one is x so the answer is l and seven this is the exact answer and then you could type it in if you want the decimal answer so the decimal answer is 1.9459 so that's the approximation now let's work on this example 5 plus 4 raised to the x minus 2 is equal to 23 let's find the value of x so let's begin by subtracting both sides by 5. we can't take the log of both sides yet it wouldn't be wise 23 minus 5 is 18. now at this point we can't really convert 18 into a base two two to the fourth is 16 2 to the fifth is 32 so we can't really change 18 to a base two so we have to take the log of both sides you can also use the natural log too it will work as well now we can move the exponent to the front and so it's x minus 2 in parentheses times log 4 and that's equal to log 18. i wouldn't recommend distributing log 4 to x minus two instead it's better to divide both sides by log four so x minus two is equal to log 18. well let's get the exact answer first so this is equal to log 18 over log 4. the exact answer is this x is equal to 2 plus log 18 over log 4. log 18 divided by log 4 that's about 2.08496 and if we add 2 to it we can see that x is equal to 4.08496 so that's the answer let's try this problem 3 plus 2 e raised to the 3 minus x let's say it's equal to 7. go ahead and find the value of x so before we take the natural log of both sides let's subtract both sides by three so two e to the three minus x is equal to four next let's divide both sides by two so e to the three minus x is equal to two and now we can take the natural log of both sides now let's move the exponent to the front so three minus x times ln e is equal to ln2 and lne is one so this is just gonna be three minus x now what i'm gonna do is take the negative x move it to this side it's negative x on the left side but it's gonna be positive x on the right side now i'm going to take ln 2 move it to this side which is going to change from positive to negative so x is 3 minus ln 2. that's the exact answer the decimal value is 2.3069 how would you solve this equation three raised to the x squared plus four let's say that's equal to one over twenty seven what would you do in order to find the value of x in this case we know that three to the third is equal to twenty seven so three to the negative three is one over twenty seven we need to make the bases the same so this is 3 to the negative 3. because the bases are the same we can now make the exponents equal to each other which means that x squared plus 4x is equal to negative 3. now let's move the negative three to the left side in which case is going to be positive three and we could factor two numbers that multiply to three but add to four are three and one so this is x plus three and x plus one so therefore x can be equal to negative three and x is equal to negative one here's another example two raised to the x squared times two raised to the three x is equal to sixteen find the value of x two to the fourth is equal to sixteen so let's replace sixteen with that now let's say if you're multiplying by a common base you can add the exponents 2 plus 3 is 5. in this case 2 is the common base so we can add x squared and three x so this is two x squared plus three x so now we can set x squared plus three x equal to four now let's subtract both sides by four so x squared plus three x minus four is equal to zero and let's factor two numbers that multiply to negative four but add to positive three are positive four and negative one so it's x plus four times x minus one which means that x is equal to negative four and it's equal to positive one try this 4 raised to the 2x minus 20 multiplied by 4 raised to the x plus 64 is equal to 0. so what can we do in order to find the value of x what i think we should do is factor but factor by substitution this equation can be reduced to a quadratic equation we're going to set a equal to 4 raised to the x which means that a squared is 4 to the 2x so let's replace 4 to the 2x with a squared and let's replace 4 to the x with a what two numbers multiply to 64 but add to negative 20. this is negative 16 and negative 4. so to factor it it's a minus 16 and a minus 4. so therefore a is equal to 16 and a is equal to 4. and we know that a is 4 to the x so therefore 4 raised to the x is equal to 16 and 4 to the x is equal to 4. 4 is basically 4 to the first power so we can clearly see that x is equal to 1. now 16 is equivalent to 4 squared so 4 to the x is equal to 4 squared which means that x is equal to 2 and so those are the two answers let's try one more example three to the two x minus three to the two x minus one let's say it's equal to eighteen go ahead and find the value of x so we can't really factor in this example because 2x is not twice the value of 2x minus 1. but we could factor the gcf we could take out a 3 to the 2x 3 to the two x divided by itself is equal to one now three to the two x minus one divided by three to the two x if we uh write it out notice what we'll get let's ignore the negative sign for now we know the overall answer is going to be negative when you divide by a common base you need to subtract the exponents 2x minus 1 minus 2x the 2x will cancel and so it's going to be 3 to the negative 1. and so that's what we have right now and if you distribute you can see that you're going to get the original equation 3 to the 2x times 1 gives you this term and 3 to the 2x times 3 to negative 1 you add the exponents 2x plus negative 1 is 2x minus 1. now let's divide both sides by this three to the negative one is basically one third so we have a complex fraction let's multiply the top and the bottom by three so eighteen times three is fifty four three times one is three three times a third is one and three minus 1 is 2 54 over 2 is 27 so 3 to the 2x is equal to 27 and 27 is 3 to the third power so 2x is equal to 3 which means that x is 3 over 2 and so that's the answer you