An Exponential System

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[scottleung]

I used logs to find a and b individually first, then used those to find the answer.

[MSJ_]

2^a = 3.
Raise both sides to power b which comes out to be 2^(ab) = 3^b.

Also, 12^b = 8 could be written as: 2^(2b)*(3^b).
Substitute (3^b) for 2^(ab),

2^(ab+2b) = 8
or,
2^(ab+2b) =2^3.

Bases are equal, hence equating the powers,
ab+2b = 3

[MrLidless]

a ln2 = ln3
b ln12 = ln8
b (2 ln2 + ln3) = 3 ln2
b (2 ln2 + a ln2) = 3 ln2
2b + ab = 3

By a parallel method;
12^b = 2^2b 3^b = 2^2b 2^ab = 2^(ab+2b) = 2^3

[ioannismichalopoulos]

You trained us so that we expect the second method to be faster and smarter than the first one. This time, faster and smarter was the first method, as it does not use properties of logarithms.

[NathanSimonGottemer]

Answer is 3.
Since 3 = 2^a and 12=3x4 we can write the second equation as 2^ab x 4^b = 8
But 4 is 2^2 so by properties of exponentials we get 2^(ab+2b)=8
Then take log base 2 of both sides and we get the answer ab+2b=3

[Paul-]

I always default to natural logs so it’s good to see other approaches.

[christianthomas]

2^a = 3 elevated power b gives: 2^(ab) = 3^b. As 12^b = 8 could be written as: 2^(2b)*(3^b) and substituting (3^b) for 2^(ab) we get:
2^(ab+2b) = 8=2^3. therefore same base is 2 and the exponants are equal: ab+2b = 3

[stevenlitvintchouk]

Simple way, also using logs to the base 2: From 2^a = 3, we get a = log 3. From 12^b = 8, we get b log 12 = log 8. Because 12 = 3 * 2^2 and 8 = 2^3, we get b*(2 + log 3) = 3. But log 3 = a. So substituting, we get b*(2+a) = 3, so 2*b + a*b = 3.

[ioannismichalopoulos]

4:15 Instead of using base 12, I would use base 2, to make simplifications simpler: b log2(12)=log2(8) <=> b=3/log2(12) <=> b= 3/(log2(3)+2)
When you plug b in together with a = logx(3) in 4:47, the simplification is obvious.

[veyselyazici]

Take ln of both sides and call x=ln3 and y=ln2. The rest is simple algebraic equation solution in one step.

[georget]

3rd method without logarithms
multiply both equations
analyze each side of the result equation into product of primes.
it will be:
2^(a+2b)×3^b = 2^3×3

thus a+2b=3 and
b=1

=> a=b=1
=> ab+2b=3

[vijaychaudhari]

Idk why when when I look power the first method come to my mind is using logs

[leickrobinson]

Just doing in my head again, but I think it’s... [SPOILER]




3? :-D

[imonkalyanbarua]

I did it using the second method. 😇 But the first one was fun too. Thank you so much. 😇🙏

[tbg-brawlstars]

I was about to find values of a and b and put it lmao 😂
Edit : I mean I was about to put in calculator 😂