Calculus Examples

$\log (\sqrt{\frac{t z}{5}})$

Step 1

Rewrite $\log (\sqrt{\frac{t z}{5}}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \sqrt{\frac{t z}{5}}$

Step 2

Solve for $t$ .

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Step 2.1

Rewrite the equation as $\sqrt{\frac{t z}{5}} = 10^{0}$ .

$\sqrt{\frac{t z}{5}} = 10^{0}$

Step 2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{t z}{5}}}^{2} = {(10^{0})}^{2}$

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{t z}{5}}$ as ${(\frac{t z}{5})}^{\frac{1}{2}}$ .

${({(\frac{t z}{5})}^{\frac{1}{2}})}^{2} = {(10^{0})}^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({(\frac{t z}{5})}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({(\frac{t z}{5})}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1.1

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

${(\frac{t z}{5})}^{\frac{1}{2} \cdot 2} = {(10^{0})}^{2}$

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

${(\frac{t z}{5})}^{\frac{1}{2} \cdot 2} = {(10^{0})}^{2}$

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(\frac{t z}{5})}^{1} = {(10^{0})}^{2}$

Step 2.3.2.1.2

Simplify.

$\frac{t z}{5} = {(10^{0})}^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify ${(10^{0})}^{2}$ .

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Step 2.3.3.1.1

Multiply the exponents in ${(10^{0})}^{2}$ .

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Step 2.3.3.1.1.1

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

$\frac{t z}{5} = 10^{0 \cdot 2}$

Step 2.3.3.1.1.2

Multiply $0$ by $2$ .

$\frac{t z}{5} = 10^{0}$

Step 2.3.3.1.2

Anything raised to $0$ is $1$ .

$\frac{t z}{5} = 1$

Step 2.4

Solve for $t$ .

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Step 2.4.1

Multiply both sides of the equation by $5$ .

$5 \frac{t z}{5} = 5 \cdot 1$

Step 2.4.2

Simplify both sides of the equation.

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Step 2.4.2.1

Simplify the left side.

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Step 2.4.2.1.1

Cancel the common factor of $5$ .

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Step 2.4.2.1.1.1

Cancel the common factor.

$5 \frac{t z}{5} = 5 \cdot 1$

Step 2.4.2.1.1.2

Rewrite the expression.

$t z = 5 \cdot 1$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Multiply $5$ by $1$ .

$t z = 5$

Step 2.4.3

Divide each term in $t z = 5$ by $z$ and simplify.

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Step 2.4.3.1

Divide each term in $t z = 5$ by $z$ .

$\frac{t z}{z} = \frac{5}{z}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $z$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{t z}{z} = \frac{5}{z}$

Step 2.4.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{5}{z}$

Step 3

List the points to graph.

Step 4

Select a few points to graph.

$\begin{array}{c} x & y \end{array}$

Step 5