Basic Math Examples

$4 \cdot 64^{n} = 1$

Step 1

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

$\ln (4 \cdot 64^{n}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (4 \cdot 64^{n})$ as $\ln (4) + \ln (64^{n})$ .

$\ln (4) + \ln (64^{n}) = \ln (1)$

Step 2.2

Expand $\ln (64^{n})$ by moving $n$ outside the logarithm.

$\ln (4) + n \ln (64) = \ln (1)$

Step 3

Simplify the right side.

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Step 3.1

The natural logarithm of $1$ is $0$ .

$\ln (4) + n \ln (64) = 0$

Step 4

Reorder $\ln (4)$ and $n \ln (64)$ .

$n \ln (64) + \ln (4) = 0$

Step 5

Subtract $\ln (4)$ from both sides of the equation.

$n \ln (64) = - \ln (4)$

Step 6

Divide each term in $n \ln (64) = - \ln (4)$ by $\ln (64)$ and simplify.

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Step 6.1

Divide each term in $n \ln (64) = - \ln (4)$ by $\ln (64)$ .

$\frac{n \ln (64)}{\ln (64)} = \frac{- \ln (4)}{\ln (64)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (64)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{n \ln (64)}{\ln (64)} = \frac{- \ln (4)}{\ln (64)}$

Step 6.2.1.2

Divide $n$ by $1$ .

$n = \frac{- \ln (4)}{\ln (64)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (4)}{\ln (64)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (4)}{\ln (64)}$

Decimal Form:

$n = - 0.\overline{3}$