How to solve exponential equations (from basic to hard!)

This algebra/precalculus tutorial covers the common exponential equation problems you must know for your algebra 2 and precalculus classes. We will focus on getting the common base, using the common logarithm and the natural log to help us solve exponential equations. We will also cover some exponential equations with different bases and factor a quadratic trinomial. Subscribe to @bprpmathbasics for more algebra tutorials.

0:00 How to solve exponential equations (from basic to difficult)
0:09 Q1. Solve 2^(3x+1)=32
1:39 Q2. Solve 27^x=9^(2x-3)
4:13 Q3. Solve 5^(x^2+3x-4)=1
6:08 Q4. Solve (sqrt(2))^(x+4)=1/8
8:37 Q5. Solve 5^x=2
12:18 Q6. Solve 20e^(3x-2)=1200
15:19 Q7. Solve 3^(x-2)=5^(x+4)
20:50 Q8. Solve 7^(2x-1)=2^(4x+3)
24:56 Q9. Solve e^(2x)+e^x-6=0
28:22 Q10. Solve 2^x+3*2^(-x)=4
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#math #algebra #mathbasics

[albertoesquivias]

Please do more of these "how to solve" type of videos and I hope it becomes a series, it was really easy to follow

[TerraBlo]

12:20
I love how he shows these random bloopers

[sankaranbs]

I thought, I have I Internet problem with 'Okay, number 6 what's e though'

[pedrogarcia]

11:15 you can also use ln(pos)/ln(b). most calculators with have natural log in addition to log base 10.

[epicstar]

for question 6 I used exponential properties before logarithmic ones

[Fenamer]

Can you make more ''How to solve' videos? I think these videos are easy to follow through, and I want more so I can learn bits of precalc and calc.

[TheZmoliver]

You take the stress out of math, man. Thank you!

[DrTWG]

You are teaching a lot of people some great maths skills .

[leonardobarrera]

Exponential equations are so easy that I have never saw on my college and I can solve all of them!!!

[barryspindler]

excellent job and done at the right pace. I like your work.

[GirishManjunathMusic]

Before watching the video:
Question 1:
2↑(3x + 1) = 32
We know that 32 = 2⁵
2↑(3x + 1) = 2⁵

Comparing like terms:
3x + 1 = 5
3x = 4
x = 0.75.

Question 2:
27↑x = 9↑(2x – 3)

27 = 3³ and 9 = 3²:
3↑3x = 3↑(4x – 6)

Comparing like:
3x = 4x – 6
x = 6.

Question 3:
5↑(x² + 3x – 4) = 1

But 5⁰ = 1:
5↑(x² + 3x – 4) = 5⁰

Comparing like:
x² + 3x – 4 = 0
x² + 4x – x – 4 = 0
(x + 4)(x – 1) = 0
x = –4, x = 1.

Question 4:
(√2)↑(x + 4) = ⅛

But ⅛ = 8↑(–1), 8 = 2² and √2 = 2↑½:
2↑(½x + 2) = 2↑(–3)

Comparing like:
½x + 2 = – 3
½x = –5
x = –10.

Question 5:
5↑x = 2
xln5 = ln2
x = ln2/ln5.

Question 6:
20e↑(3x – 2) = 1200
e↑(3x – 2) = 60
3x – 2 = ln(60)
x = ⅓(ln(60) + 2).

Question 7:
3↑(x – 2) = 5↑(x + 4)
(x – 2)ln3 = (x + 4)ln5
xln3 – 2ln3 = xln5 + 4ln5
x(ln3 – ln5) = 2ln3 + 4ln5
x = (2ln(3) + 4ln(5))/(ln(3) – ln(5)).

Question 8:
7↑(2x – 1) = 2↑(4x + 3)
(2x – 1)ln7 = (4x + 3)ln2
2xln7 – ln7 = 4xln2 + 3ln2
x(2ln7 – 4ln2) = (3ln2 + ln7)
x = (3ln2 + ln7)/(2ln7 – 4ln2).

Question 9:
e↑2x + e↑x – 6 = 0:

Setting e↑x = t → e↑2x = t²:
t² + t – 6 = 0
t² + 3t – 2t – 6 = 0
(t + 3)(t – 2) = 0
t = –3 or t = 2

But as t = e↑x; t > 0:
t = 2 → e↑x = 2
x = ln2.

Question 10:
2↑x + 3·2↑(–x) = 4

Setting 2↑x = t → 2↑(–x) = 1/t:
t + 3/t = 4
t – 4t + 3 = 0
t – 3t – t + 3 = 0
(t – 1)(t – 3) = 0
t = 1 or t = 3.

But t = 2↑x:
Case 1: t = 1:
2↑x = 1
x = 0.

Case 2: t = 3:
2↑x = 3
xln2 = ln3
x = ln(3)/ln(2).

[gnlzpnd]

Question 7:
3^(x-2) = 5^(x+4)
(3/5)^x = 5^4 * 3^2
x = log(5^4 * 3^2)/log(3/5)

Question 8:
7^(2x-1) = 2^(4x+3)
(7/4)^(2x) = 2^3 * 7
2x = log(2^3 * 7)/log(7/4)

[Gremriel]

Most of what you talk about goes over my head, but occasionally I learn something.

[klmcwhirter]

Love the video.

But I saw you miss this a couple of times now.

The reason the exponential property of same bases works is because if you take the log of both sides they cancel. _Note I saw you covered this later, but it would be helpful to explain in problem 1._

2^(3X+1) = 2^5

log₂ 2^(3X+1) = log₂ 2^5 (how do I do strikeout in a comment?)

Which leaves you with ...
3X+1 = 5

Heads up - that extra explanation will make things clearer for some students.

Great job though. I love your videos!

[fromscratch]

For 9: e^2x + e^x - 6 = 0
why wouldn't collecting like terms and using ln on both sides work?

[cdkw]

Where is the 25 find the range of function questions? You said you would upload that in your 25 trig questions video.

[axn_anime]

Can you please do one for trigonometry 🥺❤️

[fullcde]

@4:46 OMG, I know that 5^0 = 1. How did I not see that I needed to use that when I tried the problem before I watched you solve

[mpcdrol]

i am promoted to class 10th 2 months ago and i solved all. I cant believe it.

[DominickRiesland]

What do you do when the teacher tells you to express the answer to #7 and #8 as a single logarithmic function? 😂