Evaluate 3 log base 9 of x^2=6 log base 9 of x

$3 \log_{9} (x^{2}) = 6 \log_{9} (x)$

Step 1

Simplify.

Simplify $3 \log_{9} (x^{2})$ by moving $3$ inside the logarithm.

$\log_{9} ({(x^{2})}^{3}) = 6 \log_{9} (x)$

Simplify $6 \log_{9} (x)$ by moving $6$ inside the logarithm.

$\log_{9} ({(x^{2})}^{3}) = \log_{9} (x^{6})$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

${(x^{2})}^{3} = x^{6}$

Step 3

Solve for $x$ .

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2 \cdot 3} = x^{6}$

Multiply $2$ by $3$ .

$x^{6} = x^{6}$

Move all terms containing $x$ to the left side of the equation.

Subtract $x^{6}$ from both sides of the equation.

$x^{6} - x^{6} = 0$

Subtract $x^{6}$ from $x^{6}$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$