Precalculus Examples

$8 (10^{3 x}) = 12$

Step 1

Divide each term in $8 \cdot 10^{3 x} = 12$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 \cdot 10^{3 x} = 12$ by $8$ .

$\frac{8 \cdot 10^{3 x}}{8} = \frac{12}{8}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 \cdot 10^{3 x}}{8} = \frac{12}{8}$

Step 1.2.1.2

Divide $10^{3 x}$ by $1$ .

$10^{3 x} = \frac{12}{8}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $12$ and $8$ .

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Step 1.3.1.1

Factor $4$ out of $12$ .

$10^{3 x} = \frac{4 (3)}{8}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $4$ out of $8$ .

$10^{3 x} = \frac{4 \cdot 3}{4 \cdot 2}$

Step 1.3.1.2.2

Cancel the common factor.

$10^{3 x} = \frac{4 \cdot 3}{4 \cdot 2}$

Step 1.3.1.2.3

Rewrite the expression.

$10^{3 x} = \frac{3}{2}$

Step 2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{3 x}) = \log (\frac{3}{2})$

Step 3

Expand the left side.

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Step 3.1

Expand $\log (10^{3 x})$ by moving $3 x$ outside the logarithm.

$3 x \log (10) = \log (\frac{3}{2})$

Step 3.2

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

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

Step 3.3

Multiply $3$ by $1$ .

$3 x = \log (\frac{3}{2})$

Step 4

Divide each term in $3 x = \log (\frac{3}{2})$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = \log (\frac{3}{2})$ by $3$ .

$\frac{3 x}{3} = \frac{\log (\frac{3}{2})}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\log (\frac{3}{2})}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\log (\frac{3}{2})}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\log (\frac{3}{2})}{3}$

Decimal Form:

$x = 0.05869708 \dots$