Precalculus Examples

$\log (5) x = 3 \log (5) \cdot 25 + 0.5 \log (5) \cdot 81 - 2 \log (5) \cdot 3$

Step 1

Simplify the right side.

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Step 1.1

Simplify $3 \log (5) \cdot 25 + 0.5 \log (5) \cdot 81 - 2 \log (5) \cdot 3$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Multiply $25$ by $3$ .

$\log (5) x = 75 \log (5) + 0.5 \log (5) \cdot 81 - 2 \log (5) \cdot 3$

Step 1.1.1.2

Multiply $81$ by $0.5$ .

$\log (5) x = 75 \log (5) + 40.5 \log (5) - 2 \log (5) \cdot 3$

Step 1.1.1.3

Multiply $3$ by $- 2$ .

$\log (5) x = 75 \log (5) + 40.5 \log (5) - 6 \log (5)$

Step 1.1.2

Simplify by adding terms.

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Step 1.1.2.1

Add $75 \log (5)$ and $40.5 \log (5)$ .

$\log (5) x = 115.5 \log (5) - 6 \log (5)$

Step 1.1.2.2

Subtract $6 \log (5)$ from $115.5 \log (5)$ .

$\log (5) x = 109.5 \log (5)$

Step 2

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

$\log (5) x - 109.5 \log (5) = 0$

Step 3

Multiply $- 109.5$ by $\log (5)$ .

$\log (5) x - 76.53721547 = 0$

Step 4

Reorder factors in $\log (5) x - 76.53721547$ .

$x \log (5) - 76.53721547 = 0$

Step 5

Add $76.53721547$ to both sides of the equation.

$x \log (5) = 76.53721547$

Step 6

Divide each term in $x \log (5) = 76.53721547$ by $\log (5)$ and simplify.

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Step 6.1

Divide each term in $x \log (5) = 76.53721547$ by $\log (5)$ .

$\frac{x \log (5)}{\log (5)} = \frac{76.53721547}{\log (5)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\log (5)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x \log (5)}{\log (5)} = \frac{76.53721547}{\log (5)}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{76.53721547}{\log (5)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Log base $10$ of $5$ is approximately $0.69897$ .

$x = \frac{76.53721547}{0.69897}$

Step 6.3.2

Divide $76.53721547$ by $0.69897$ .

$x = 109.5$