Solve for x log of 20x^3-2 log of x=4

$\log (20 x^{3}) - 2 \log (x) = 4$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log (20 x^{3}) - 2 \log (x) = 4$

Step 2

Simplify the left side.

Simplify $\log (20 x^{3}) - 2 \log (x)$ .

Simplify $- 2 \log (x)$ by moving $2$ inside the logarithm.

$\log (20 x^{3}) - \log (x^{2}) = 4$

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

$\log (\frac{20 x^{3}}{x^{2}}) = 4$

Cancel the common factor of $x^{3}$ and $x^{2}$ .

Factor $x^{2}$ out of $20 x^{3}$ .

$\log (\frac{x^{2} (20 x)}{x^{2}}) = 4$

Cancel the common factors.

Multiply by $1$ .

$\log (\frac{x^{2} (20 x)}{x^{2} \cdot 1}) = 4$

Cancel the common factor.

$\log (\frac{x^{2} (20 x)}{x^{2} \cdot 1}) = 4$

Rewrite the expression.

$\log (\frac{20 x}{1}) = 4$

Divide $20 x$ by $1$ .

$\log (20 x) = 4$

Step 3

Rewrite $\log (20 x) = 4$ 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 $b^{y} = x$ .

$10^{4} = 20 x$

Step 4

Solve for $x$ .

Rewrite the equation as $20 x = 10^{4}$ .

$20 x = 10^{4}$

Divide each term in $20 x = 10^{4}$ by $20$ and simplify.

Divide each term in $20 x = 10^{4}$ by $20$ .

$\frac{20 x}{20} = \frac{10^{4}}{20}$

Simplify the left side.

Cancel the common factor of $20$ .

Cancel the common factor.

$\frac{20 x}{20} = \frac{10^{4}}{20}$

Divide $x$ by $1$ .

$x = \frac{10^{4}}{20}$

Simplify the right side.

Raise $10$ to the power of $4$ .

$x = \frac{10000}{20}$

Divide $10000$ by $20$ .

$x = 500$

Solve for x log of 20x^3-2 log of x=4

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