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### Course: Algebra (all content)>Unit 11

Lesson 16: Solving exponential equations using properties of exponents

# Solving exponential equations using exponent properties

Sal solves equations like 26^(9x+5) = 1 and 2^(3x+5) = 64^(x-7).

## Want to join the conversation?

• wait... so in the first equation he divided -5 by 9 and kept -5 instead of making it.5 repeating. why?
• This is a known error in the video. If you watch the video in normal mode (not full screen), you would see a correction box pop-up and tell you that Sal meant x=-5/9
• how would i solve 2^-2b = 2^-b
• For the 2 sides of your equation to be equal, the exponents must be equal.
So, you can change the equation into: ` -2b = -b`
Then, solve for "b"
Sal does something very similar at about in the video.
Hope this helps.
• What is the operation called when you turn a fraction into a multiple of some number...? Like `1/32` becomes `2^-5` ! And is there any way to perform this sort of alteration easily!?^_^"

• You just rewrote the number in its factored form using prime factors and exponents.
If you know how to do prime factorization and division, breaking the number down is not hard.
• I can't believe this... not only do we have normal equations...but we now have equations on the exponents... this is pretty neat, it's like we're "zooming in" on the base number's exponents who have their own "world" of math going on, right above the base numbers that we normally work with. I do have a question though. For the first problem with the 26=1...couldn't you raise the 26 to the zero power...then have that 0(9x+5) multiply out and equal 0...then it would leave 26^0 which is 1...and that would equal 1=1? This isn't how Sal does it, but I think it's a valid method?
• That is a valid operation that results in a true statement, but it doesn't actually help us find x. We can also multiply both sides of an equation by 0, but that's not helpful either.
• *Didn't he do the first one wrong?*
• This is a known error in the video. A correction box appears at @ in the video. Note: you will only see the correction boxes if you watch in regular mode (not full screen mode). So, always check for a correction box before posting that you think there is an error.

The correct result for the 1st problem is x = -5/9
• Before seeing the obvious strategy in the first example is to use the x^0 = 1 I calculate the answer to be x = -5/4. I wonder if somebody could help identify where my logic is incorrect:

26^9x+5 = 26^9x `*` 26^5 (split off +5)

(26^9)^x `*` 26^5 (factor out ^x)

(26^5 * 26^4)^x `*` 26^5 (rewrite 26^9 as product of ^5+4)

26^5(26^4)^x (factor out 26^5*)

26^4x = 1/26^5 (divide by 26^5)

26^4x = 26^-5 (reciprocate the 1/26^5)

4x = -5 (solve for x)

*Is it that I factor out incorrectly here? I cant factor out one 26^5 as there is essentially ^x amount of 26^5's so can't reduce it out as just one?

Many Thanks to whichever kind stranger knows!
• One problem is is in step 4, you cannot factor out things using multiplication, factoring out is used with adding terms. Lets use simpler terms (2^2*2^3)^3*2^2 = (4*8)^3*4 = 131072. However, 2^2(2^3)^3=2048, so these two are clearly not the same. You would have (26^5)^(x+1)*26^4 which is not getting you any closer to the answer. I think you realized the issue of factoring by your question near the end.
• How would you solve
3^(x+3)-3^(x)=78?
• I can only get to 3^x = 78/8.
(1 vote)
• in the first one isnt x= -5/9?
• Yes. This is a known error in the video. A correction box pops up at about in the video and tells you Sal meant to write x=-5/9.
• At , I don't get why x^a=x^b proves that a=b. Can someone please give me justification on why that is the case?