Solve for x log of 5x^3- log of x^2=2

$\log (5 x^{3}) - \log (x^{2}) = 2$

Step 1

Simplify the left side.

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

$\log (\frac{5 x^{3}}{x^{2}}) = 2$

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

Factor $x^{2}$ out of $5 x^{3}$ .

$\log (\frac{x^{2} (5 x)}{x^{2}}) = 2$

Cancel the common factors.

Multiply by $1$ .

$\log (\frac{x^{2} (5 x)}{x^{2} \cdot 1}) = 2$

Cancel the common factor.

$\log (\frac{x^{2} (5 x)}{x^{2} \cdot 1}) = 2$

Rewrite the expression.

$\log (\frac{5 x}{1}) = 2$

Divide $5 x$ by $1$ .

$\log (5 x) = 2$

Step 2

Rewrite $\log (5 x) = 2$ 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 $b^{y} = x$ .

$10^{2} = 5 x$

Step 3

Solve for $x$ .

Rewrite the equation as $5 x = 10^{2}$ .

$5 x = 10^{2}$

Divide each term in $5 x = 10^{2}$ by $5$ and simplify.

Divide each term in $5 x = 10^{2}$ by $5$ .

$\frac{5 x}{5} = \frac{10^{2}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{10^{2}}{5}$

Divide $x$ by $1$ .

$x = \frac{10^{2}}{5}$

Simplify the right side.

Raise $10$ to the power of $2$ .

$x = \frac{100}{5}$

Divide $100$ by $5$ .

$x = 20$

Solve for x log of 5x^3- log of x^2=2

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