Need help with solving equation

[fawazalusail]

Solve the equation.

3^(2x+1)=4^x
-----------------------------

ln(3^(2x+1)=ln(4^x)
(2x+1)ln(3)=xln(4)
2xln(3)+ln(3)=xln(4)
2xln(3)=xln(4)-ln(3)
2xln(3)-xln(4)=-ln(3)
x(2ln(3)-ln(4))=-ln(3)

The answer is, x= -1.3548

What i don't understand is how did we get that result ^^ i tried to divide but couldn't get the same answer ??

Please help. Thanks

 

[mmm4444bot]

Solve the equation.

3^(2x+1) = 4^x

-----------------------------

ln(3^(2x+1)) = ln(4^x)

(2x+1)ln(3) = xln(4)

2xln(3) + ln(3) = xln(4)

2xln(3) = xln(4) - ln(3)

2xln(3) - xln(4) = -ln(3)

x(2ln(3) - ln(4)) = -ln(3)

This is very good work, so far. You need to simplify some stuff.

You have the expression 2*ln(3) above. This is the same as ln(3^2), yes?

So, your last line above simplifies to:

x[ln(9) - ln(4)] = -ln(3)

Next, you can simplify ln(9)-ln(4). Do you know the property?

Last step is to divide both sides of equation to isolate x.

Can you finish now? :cool:

 

[fawazalusail]

I'm sorry, I still don't get it. can you be more specific please.

 

[mmm4444bot]

I still don't get it. can you be more specific please.

What does your pronoun it represent? In other words, what are you trying to ask about? Can you be more specific, please? :cool:

Have you seen the following property?

ln(a) - ln(b) = ln(a/b)

 

[JeffM]

I'm sorry, I still don't get it. can you be more specific please.

\(\displaystyle x\{ln(9) - ln(4)\} = -ln(3) \implies\)

\(\displaystyle x * ln\left(\dfrac{9}{4}\right) = -ln(3) \implies\)

\(\displaystyle x * ln(2.25) = - ln(3)\)

\(\displaystyle x = \dfrac{- ln(3) }{ln(2.25)} \approx -1.3548.\)

Let's check


\(\displaystyle 3^{[2(-1.3548) + 1]} = 3^{(-1.7096)} \approx 0.1529.\)

\(\displaystyle 4^{(-1.3548)} \approx 0.1529.\)

 

[fawazalusail]

\(\displaystyle x\{ln(9) - ln(4)\} = -ln(3) \implies\)

\(\displaystyle x * ln\left(\dfrac{9}{4}\right) = -ln(3) \implies\)

\(\displaystyle x * ln(2.25) = - ln(3)\)

\(\displaystyle x = \dfrac{- ln(3) }{ln(2.25)} \approx -1.3548.\)

Let's check


\(\displaystyle 3^{[2(-1.3548) + 1]} = 3^{(-1.7096)} \approx 0.1529.\)

\(\displaystyle 4^{(-1.3548)} \approx 0.1529.\)

Thanks JeffM now I get it.