Solve for x log base 4 of 9x- log base 4 of 3=0

$\log_{4} (9 x) - \log_{4} (3) = 0$

Step 1

Simplify the left side.

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{4} (\frac{9 x}{3}) = 0$

Cancel the common factor of $9$ and $3$ .

Factor $3$ out of $9 x$ .

$\log_{4} (\frac{3 (3 x)}{3}) = 0$

Cancel the common factors.

Factor $3$ out of $3$ .

$\log_{4} (\frac{3 (3 x)}{3 (1)}) = 0$

Cancel the common factor.

$\log_{4} (\frac{3 (3 x)}{3 \cdot 1}) = 0$

Rewrite the expression.

$\log_{4} (\frac{3 x}{1}) = 0$

Divide $3 x$ by $1$ .

$\log_{4} (3 x) = 0$

Step 2

Rewrite $\log_{4} (3 x) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$4^{0} = 3 x$

Step 3

Solve for $x$ .

Rewrite the equation as $3 x = 4^{0}$ .

$3 x = 4^{0}$

Anything raised to $0$ is $1$ .

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}$

Decimal Form:

$x = 0.\overline{3}$

Solve for x log base 4 of 9x- log base 4 of 3=0

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