Solve for $n$ in $18^{n+1} = 2^{n+1} \cdot 27$

Question

Solve for $n$: $$18^{n+1} = 2^{n+1} \cdot 27$$

I tried:

$$18^{n+1} = 2^{n+1} \cdot 27 \Leftrightarrow 18^n \cdot 18 = 2^n \cdot 54 \Leftrightarrow \frac{18^n}{54} = \frac{2^n}{18} \Leftrightarrow \frac{18 \cdot 18^n - 54 \cdot 2^n}{972} = 0 \Leftrightarrow \\ 18 \cdot 18^n - 54 \cdot 2^n = 0 \Leftrightarrow ???$$

What do I do next? Am I doing it right?

Answer 1

Divide both sides by $2^{n+1}$.

$$9^{n+1}=27$$

$$n+1=\log_9(27)=\frac 32$$

$$n=\frac 1 2$$

Answer 2

The problem with what you did

What you have done so far are valid algebraic manipulations, but you haven't made any progress. You ended up with the equation $$ 18 \cdot 18^n - 54 \cdot 2^n = 0 $$ which you will notice is the exact same equation you had in the second step, $18 \cdot 18^n = 54 \cdot 2^n$, so all those intermediate steps (where you subtracted and found a common denominator) didn't get anywhere.

Extended hint

To get somewhere (make progress), you want to isolate $n$ which means bring all the things with $n$ to one side. So you have $$ 18 \cdot 18^n = 54 \cdot 2^n $$ and you want to bring all the $n$s to one side, so you can divide by $2^n$: $$ \frac{18^n}{2^n} = \frac{54}{18} $$ I will leave it to you to simplify $\frac{54}{18}$. What about the other side? Well, $$ \frac{18^n}{2^n} = \left( \frac{18}{2} \right)^n = ? $$

Answer 3

$18^{n+1}=2^{n+1}⋅27$
$\implies (3^{2}⋅2)^{n+1}=2^{n+1}⋅3^{3}$
$\implies 3^{2(n+1)}⋅2^{n+1}=2^{n+1}⋅3^{3}$
$\implies 3^{2(n+1)}=3^{3}$
$\implies 2(n+1) = 3$

Answer 4

$$18^{n+1} = 2^{n+1}\cdot27$$ $$\Leftrightarrow \frac{18^{n+1}}{2^{n+1}} = 27$$ $$\Leftrightarrow \left(\frac{18}{2}\right)^{n+1} = 27$$ $$\Leftrightarrow 9^{n+1} = 27$$ $$\Leftrightarrow 9^{n} = 3$$ $$\Leftrightarrow n =\log_9(3) =\frac {\log 3}{\log 9} =\frac {\log 3}{2\cdot \log 3} = \frac 1 2$$

Answer 5

Since $18=9\cdot 2$, we can rewrite the equation as$$(9\cdot2)^{n+1}=2^{n+1}\cdot 27\tag1$$ Simplifying $(1)$, we can eliminate $2^{n+1}$ to obtain$$3^{2n+2}=3^3\tag2$$ Which can be solved by $2n+2=3$. Solving, we get $n=\frac 12$.

Answer 6

$18^{n+1}=2^{n+1}\times 27$

$3^{2n+2}\times 2^{n+1}=2^{n+1}\times3^{3}$

$3^{2n+2}=3^3$

$2n+2=3$

$n=1/2$

Answer 7

Well, your not doing it wrong.

For better or worse, you reached:

$18*18^n -2^n*54=0$

Continue:

$2^n (18*9^n-54)=0$

$9*9^n-27=0$

$9^n-3=0$

$3^{2n}=3^1$

$2n=1$

$n=1/2$

So keep going and you'll get an answer.

But once done and in hindsight, it might become apparent although you were doing it a not wrong way, you weren't necessarily doing it the "right" way.