Algebra Examples

${(0.5)}^{\frac{9900}{x}} = 0.3$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({0.5}^{\frac{9900}{x}}) = \ln (0.3)$

Step 2

Expand the left side.

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Step 2.1

Expand

$\ln ({0.5}^{\frac{9900}{x}})$ by moving

$\frac{9900}{x}$ outside the logarithm.

$\frac{9900}{x} \ln (0.5) = \ln (0.3)$

Step 2.2

Combine

$\frac{9900}{x}$ and

$\ln (0.5)$ .

$\frac{9900 \ln (0.5)}{x} = \ln (0.3)$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x$

Step 4

Multiply each term in

$\frac{9900 \ln (0.5)}{x} = \ln (0.3)$ by

$x$ to eliminate the fractions.

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Step 4.1

Multiply each term in

$\frac{9900 \ln (0.5)}{x} = \ln (0.3)$ by

$x$ .

$\frac{9900 \ln (0.5)}{x} x = \ln (0.3) x$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

$x$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{9900 \ln (0.5)}{x} x = \ln (0.3) x$

Step 4.2.1.2

Rewrite the expression.

$9900 \ln (0.5) = \ln (0.3) x$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as

$\ln (0.3) x = 9900 \ln (0.5)$ .

$\ln (0.3) x = 9900 \ln (0.5)$

Step 5.2

Divide each term in

$\ln (0.3) x = 9900 \ln (0.5)$ by

$\ln (0.3)$ and simplify.

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Step 5.2.1

Divide each term in

$\ln (0.3) x = 9900 \ln (0.5)$ by

$\ln (0.3)$ .

$\frac{\ln (0.3) x}{\ln (0.3)} = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$\ln (0.3)$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{\ln (0.3) x}{\ln (0.3)} = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Decimal Form:

$x = 5699.59476068 \dots$