Pre-Algebra Examples

$4^{- 3 t} = 0.1$

Step 1

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

$\ln (4^{- 3 t}) = \ln (0.1)$

Step 2

Expand $\ln (4^{- 3 t})$ by moving $- 3 t$ outside the logarithm.

$- 3 t \ln (4) = \ln (0.1)$

Step 3

Divide each term in $- 3 t \ln (4) = \ln (0.1)$ by $- 3 \ln (4)$ and simplify.

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Step 3.1

Divide each term in $- 3 t \ln (4) = \ln (0.1)$ by $- 3 \ln (4)$ .

$\frac{- 3 t \ln (4)}{- 3 \ln (4)} = \frac{\ln (0.1)}{- 3 \ln (4)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 3 t \ln (4)}{- 3 \ln (4)} = \frac{\ln (0.1)}{- 3 \ln (4)}$

Step 3.2.1.2

Rewrite the expression.

$\frac{t \ln (4)}{\ln (4)} = \frac{\ln (0.1)}{- 3 \ln (4)}$

Step 3.2.2

Cancel the common factor of $\ln (4)$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{t \ln (4)}{\ln (4)} = \frac{\ln (0.1)}{- 3 \ln (4)}$

Step 3.2.2.2

Divide $t$ by $1$ .

$t = \frac{\ln (0.1)}{- 3 \ln (4)}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (0.1)}{3 \ln (4)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{\ln (0.1)}{3 \ln (4)}$

Decimal Form:

$t = 0.55365468 \dots$