Precalculus Examples

$8 \cdot 5^{- \frac{t}{9}} = 346$

Step 1

Divide each term in $8 \cdot 5^{- \frac{t}{9}} = 346$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 \cdot 5^{- \frac{t}{9}} = 346$ by $8$ .

$\frac{8 \cdot 5^{- \frac{t}{9}}}{8} = \frac{346}{8}$

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.

$\frac{8 \cdot 5^{- \frac{t}{9}}}{8} = \frac{346}{8}$

Step 1.2.1.2

Divide $5^{- \frac{t}{9}}$ by $1$ .

$5^{- \frac{t}{9}} = \frac{346}{8}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $346$ and $8$ .

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Step 1.3.1.1

Factor $2$ out of $346$ .

$5^{- \frac{t}{9}} = \frac{2 (173)}{8}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $8$ .

$5^{- \frac{t}{9}} = \frac{2 \cdot 173}{2 \cdot 4}$

Step 1.3.1.2.2

Cancel the common factor.

$5^{- \frac{t}{9}} = \frac{2 \cdot 173}{2 \cdot 4}$

Step 1.3.1.2.3

Rewrite the expression.

$5^{- \frac{t}{9}} = \frac{173}{4}$

Step 2

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

$\ln (5^{- \frac{t}{9}}) = \ln (\frac{173}{4})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (5^{- \frac{t}{9}})$ by moving $- \frac{t}{9}$ outside the logarithm.

$- \frac{t}{9} \ln (5) = \ln (\frac{173}{4})$

Step 3.2

Combine $\ln (5)$ and $\frac{t}{9}$ .

$- \frac{\ln (5) t}{9} = \ln (\frac{173}{4})$

Step 4

Simplify the left side.

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Step 4.1

Reorder factors in $- \frac{\ln (5) t}{9}$ .

$- \frac{t \ln (5)}{9} = \ln (\frac{173}{4})$

Step 5

Multiply both sides of the equation by $- \frac{9}{\ln (5)}$ .

$- \frac{9}{\ln (5)} (- \frac{t \ln (5)}{9}) = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- \frac{9}{\ln (5)} (- \frac{t \ln (5)}{9})$ .

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Step 6.1.1.1

Cancel the common factor of $9$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{9}{\ln (5)}$ into the numerator.

$\frac{- 9}{\ln (5)} (- \frac{t \ln (5)}{9}) = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.1.2

Move the leading negative in $- \frac{t \ln (5)}{9}$ into the numerator.

$\frac{- 9}{\ln (5)} \cdot \frac{- t \ln (5)}{9} = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.1.3

Factor $9$ out of $- 9$ .

$\frac{9 (- 1)}{\ln (5)} \cdot \frac{- t \ln (5)}{9} = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{9 \cdot - 1}{\ln (5)} \cdot \frac{- t \ln (5)}{9} = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{\ln (5)} (- t \ln (5)) = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.2

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

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Step 6.1.1.2.1

Factor $\ln (5)$ out of $- t \ln (5)$ .

$\frac{- 1}{\ln (5)} (\ln (5) (- t)) = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{\ln (5)} (\ln (5) (- t)) = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.2.3

Rewrite the expression.

$- - t = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 t = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.1.1.3.2

Multiply $t$ by $1$ .

$t = - \frac{9}{\ln (5)} \ln (\frac{173}{4})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{9}{\ln (5)} \ln (\frac{173}{4})$ .

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Step 6.2.1.1

Combine $\ln (\frac{173}{4})$ and $\frac{9}{\ln (5)}$ .

$t = - \frac{\ln (\frac{173}{4}) \cdot 9}{\ln (5)}$

Step 6.2.1.2

Move $9$ to the left of $\ln (\frac{173}{4})$ .

$t = - \frac{9 \ln (\frac{173}{4})}{\ln (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{9 \ln (\frac{173}{4})}{\ln (5)}$

Decimal Form:

$t = - 21.06510281 \dots$