Need help asap. Thank you!

Question

1. Solve the equation 4^2x = 2^{x - 6}. Explain.

2. Solve the equation log₅(6x - 1) = 3. Explain.

3. Expand the expression . Explain.

Answer 1

1. Solve the equation 4^2x = 2^{x - 6}.

2^4x = 2^x-6, from here 4x = x - 6; 3x = - 6; x = - 2.

2. Solve the equation log₅(6x - 1) = 3; 6x - 1 = 5³; 6x = 126; x = 21.

3.Expand the expression = ln2 + 3lnx - ln5 - lny

Answer 2

4^2x = 2^4x set the exponents equal, solve for x

4x = x-6

3x =-6

x=-6/ =-2

2 5^3 = 125 =6x-1

6x = 126

x = 126/6 = 21

hard to read the 3rd one. I don't have magnifying glass or microscope. I'm up to 3.5 magnification on reading glasses. so far.

maybe it's log(2x^3/3y) if so then

=log2x^3 -log3y = log2 + 3logx- log3y = log2 + 3logx -log3 -logy

= 3logx -logy + log(2/3) = 3logx -logy -4.641588834 if log is base 10, which it usually is, at least in less advanced math courses. later log = ln with base e IF so, then log(2/3 = -.4054651 approximately

some log rules are log(a/b) = loga - logb

and loga^b = bloga , exponent comes out as a coefficient

Answer 3

(1)

Mr. Yefim's answer is much quicker in this instance. However, the bases may NOT always be compatible.

That is you may not be able to rewrite the equation so that both sides have the common base.

For example base 8 vs. base 6

If that happens, you need to learn how to do it this way....

First, Takes log of both sides. Then by properties of LOG:

2x* (log 4) = (x-6) (log 2)

Let A=log4 and B=log2:

2A x = B(x-6)

2Ax =Bx-6b

2Ax-Bx =-6b

x(2A-B) =-6b

x =(-6b)/(2A-B)

approximation left for you , where A=log4 and b=log2 but it is really

-6 log2 / (2 log 4 - log 2) =

-6 log 2 / (2 log 2^2 -log2)=

-6 log 2/(4log2-log2)=

-6 log2/(3log2)=

the log2 cancels leaving the answer -2 which agrees with Mr.Yefim's answer.

(Raymond, thank you but you divided by 2 instead of 3).

Thanks Kinny!

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(2)

The per base 5:

6x-1 = 5^3

6x-1 = 125

6x = 126

x=126/6 =21

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(3) one log per var:

ln (2x^3) - ln(5y) =

ln2 + lnx^3 - [ ln5 + lny]=

ln2 + 3lnx -ln5 -lny