Calculus Examples

$\frac{9}{8} \cdot \log_{5} (x) = 4$

Step 1

Multiply both sides of the equation by $\frac{8}{9}$ .

$\frac{8}{9} (\frac{9}{8} \cdot \log_{5} (x)) = \frac{8}{9} \cdot 4$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{8}{9} (\frac{9}{8} \cdot \log_{5} (x))$ .

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Step 2.1.1.1

Combine $\frac{9}{8}$ and $\log_{5} (x)$ .

$\frac{8}{9} \cdot \frac{9 \log_{5} (x)}{8} = \frac{8}{9} \cdot 4$

Step 2.1.1.2

Cancel the common factor of $8$ .

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Step 2.1.1.2.1

Cancel the common factor.

$\frac{8}{9} \cdot \frac{9 \log_{5} (x)}{8} = \frac{8}{9} \cdot 4$

Step 2.1.1.2.2

Rewrite the expression.

$\frac{1}{9} (9 \log_{5} (x)) = \frac{8}{9} \cdot 4$

Step 2.1.1.3

Cancel the common factor of $9$ .

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Step 2.1.1.3.1

Factor $9$ out of $9 \log_{5} (x)$ .

$\frac{1}{9} (9 (\log_{5} (x))) = \frac{8}{9} \cdot 4$

Step 2.1.1.3.2

Cancel the common factor.

$\frac{1}{9} (9 \log_{5} (x)) = \frac{8}{9} \cdot 4$

Step 2.1.1.3.3

Rewrite the expression.

$\log_{5} (x) = \frac{8}{9} \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $\frac{8}{9} \cdot 4$ .

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Step 2.2.1.1

Combine $\frac{8}{9}$ and $4$ .

$\log_{5} (x) = \frac{8 \cdot 4}{9}$

Step 2.2.1.2

Multiply $8$ by $4$ .

$\log_{5} (x) = \frac{32}{9}$

Step 3

Rewrite $\log_{5} (x) = \frac{32}{9}$ 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 $b^{y} = x$ .

$5^{\frac{32}{9}} = x$

Step 4

Rewrite the equation as $x = 5^{\frac{32}{9}}$ .

$x = 5^{\frac{32}{9}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 5^{\frac{32}{9}}$

Decimal Form:

$x = 305.65160601 \dots$