Graph f(x)=1/3 log of 5x

$f (x) = \frac{1}{3} \log (5 x)$

Step 1

Find the asymptotes.

Set the argument of the logarithm equal to zero.

$5^{\frac{1}{3}} x^{\frac{1}{3}} = 0$

Solve for $x$ .

Divide each term in $5^{\frac{1}{3}} x^{\frac{1}{3}} = 0$ by $5^{\frac{1}{3}}$ and simplify.

Divide each term in $5^{\frac{1}{3}} x^{\frac{1}{3}} = 0$ by $5^{\frac{1}{3}}$ .

$\frac{5^{\frac{1}{3}} x^{\frac{1}{3}}}{5^{\frac{1}{3}}} = \frac{0}{5^{\frac{1}{3}}}$

Simplify the left side.

Cancel the common factor.

$\frac{5^{\frac{1}{3}} x^{\frac{1}{3}}}{5^{\frac{1}{3}}} = \frac{0}{5^{\frac{1}{3}}}$

Divide $x^{\frac{1}{3}}$ by $1$ .

$x^{\frac{1}{3}} = \frac{0}{5^{\frac{1}{3}}}$

Simplify the right side.

Divide $0$ by $5^{\frac{1}{3}}$ .

$x^{\frac{1}{3}} = 0$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Rewrite the expression.

$x^{1} = 0^{3}$

Simplify.

$x = 0^{3}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x = 0$

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{1}{3} \cdot \log (5 (1))$

Simplify the result.

Multiply $5$ by $1$ .

$f (1) = \frac{1}{3} \cdot \log (5)$

Simplify $\frac{1}{3} \log (5)$ by moving $\frac{1}{3}$ inside the logarithm.

$f (1) = \log (5^{\frac{1}{3}})$

The final answer is $\log (5^{\frac{1}{3}})$ .

$\log (5^{\frac{1}{3}})$

Convert $\log (5^{\frac{1}{3}})$ to decimal.

$y = 0.23299$

Step 3

Find the point at $x = 2$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{1}{3} \cdot \log (5 (2))$

Simplify the result.

Multiply $5$ by $2$ .

$f (2) = \frac{1}{3} \cdot \log (10)$

Logarithm base $10$ of $10$ is $1$ .

$f (2) = \frac{1}{3} \cdot 1$

Multiply $\frac{1}{3}$ by $1$ .

$f (2) = \frac{1}{3}$

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Convert $\frac{1}{3}$ to decimal.

$y = 0.\overline{3}$

Step 4

Find the point at $x = 3$ .

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{1}{3} \cdot \log (5 (3))$

Simplify the result.

Multiply $5$ by $3$ .

$f (3) = \frac{1}{3} \cdot \log (15)$

Simplify $\frac{1}{3} \log (15)$ by moving $\frac{1}{3}$ inside the logarithm.

$f (3) = \log (15^{\frac{1}{3}})$

The final answer is $\log (15^{\frac{1}{3}})$ .

$\log (15^{\frac{1}{3}})$

Convert $\log (15^{\frac{1}{3}})$ to decimal.

$y = 0.39203041$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0.23299), (2, 0.\overline{3}), (3, 0.39203041)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0.233 \\ 2 & 0.333 \\ 3 & 0.392 \end{array}$

Step 6

Graph f(x)=1/3 log of 5x

Related Questions

Support LCMGCF.com by disabling your adblocker.