Precalculus Examples

$3 \log (4 x) - 2 = 10$

Step 1

Move all terms not containing $\log (4 x)$ to the right side of the equation.

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Step 1.1

Add $2$ to both sides of the equation.

$3 \log (4 x) = 10 + 2$

Step 1.2

Add $10$ and $2$ .

$3 \log (4 x) = 12$

Step 2

Divide each term in $3 \log (4 x) = 12$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 \log (4 x) = 12$ by $3$ .

$\frac{3 \log (4 x)}{3} = \frac{12}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 \log (4 x)}{3} = \frac{12}{3}$

Step 2.2.1.2

Divide $\log (4 x)$ by $1$ .

$\log (4 x) = \frac{12}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $12$ by $3$ .

$\log (4 x) = 4$

Step 3

Rewrite $\log (4 x) = 4$ 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 $b^{y} = x$ .

$10^{4} = 4 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $4 x = 10^{4}$ .

$4 x = 10^{4}$

Step 4.2

Divide each term in $4 x = 10^{4}$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 x = 10^{4}$ by $4$ .

$\frac{4 x}{4} = \frac{10^{4}}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{10^{4}}{4}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{4}}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $10$ to the power of $4$ .

$x = \frac{10000}{4}$

Step 4.2.3.2

Divide $10000$ by $4$ .

$x = 2500$