Solve the following equation for $$ x $$. $$ \log _{27} x+\log _{27}(x+8)=\frac{2}{3} $$ $$ x=\square $$

Question

UpStudy AI · Accepted Answer

Solve the equation $$ \log_{27}x+\log_{27}(x+8)=\frac{2}{3} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{27}{\left(xight)}+\log_{27}{\left(x+8ight)}=\frac{2}{3}$$
- step1: Find the domain:
  $$\log_{27}{\left(xight)}+\log_{27}{\left(x+8ight)}=\frac{2}{3},x>0$$
- step2: Transform the expression:
  $$\log_{27}{\left(x\left(x+8ight)ight)}=\frac{2}{3}$$
- step3: Convert the logarithm into exponential form:
  $$x\left(x+8ight)=27^{\frac{2}{3}}$$
- step4: Evaluate the power:
  $$x\left(x+8ight)=9$$
- step5: Expand the expression:
  $$x^{2}+8x=9$$
- step6: Move the expression to the left side:
  $$x^{2}+8x-9=0$$
- step7: Factor the expression:
  $$\left(x-1ight)\left(x+9ight)=0$$
- step8: Separate into possible cases:
  $$\begin{align}&x-1=0\&x+9=0\end{align}$$
- step9: Solve the equation:
  $$\begin{align}&x=1\&x=-9\end{align}$$
- step10: Check if the solution is in the defined range:
  $$\begin{align}&x=1\&x=-9\end{align},x>0$$
- step11: Find the intersection:
  $$x=1$$
The solution to the equation $$ \log _{27} x+\log _{27}(x+8)=\frac{2}{3} $$ is $$ x=1 $$.
$$ x=1 $$.

Solve the following equation for .

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions