Finite Math Examples

$2 \cdot 5^{- X} = 18$

Step 1

Divide each term in $2 \cdot 5^{- X} = 18$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \cdot 5^{- X} = 18$ by $2$ .

$\frac{2 \cdot 5^{- X}}{2} = \frac{18}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cdot 5^{- X}}{2} = \frac{18}{2}$

Step 1.2.1.2

Divide $5^{- X}$ by $1$ .

$5^{- X} = \frac{18}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $18$ by $2$ .

$5^{- X} = 9$

Step 2

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

$\ln (5^{- X}) = \ln (9)$

Step 3

Expand $\ln (5^{- X})$ by moving $- X$ outside the logarithm.

$- X \ln (5) = \ln (9)$

Step 4

Divide each term in $- X \ln (5) = \ln (9)$ by $- \ln (5)$ and simplify.

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Step 4.1

Divide each term in $- X \ln (5) = \ln (9)$ by $- \ln (5)$ .

$\frac{- X \ln (5)}{- \ln (5)} = \frac{\ln (9)}{- \ln (5)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{X \ln (5)}{\ln (5)} = \frac{\ln (9)}{- \ln (5)}$

Step 4.2.2

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

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Step 4.2.2.1

Cancel the common factor.

$\frac{X \ln (5)}{\ln (5)} = \frac{\ln (9)}{- \ln (5)}$

Step 4.2.2.2

Divide $X$ by $1$ .

$X = \frac{\ln (9)}{- \ln (5)}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$X = - \frac{\ln (9)}{\ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$X = - \frac{\ln (9)}{\ln (5)}$

Decimal Form:

$X = - 1.36521238 \dots$