Basic Math Examples

$4^{n - 2} = 3$

Step 1

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

$\ln (4^{n - 2}) = \ln (3)$

Step 2

Expand $\ln (4^{n - 2})$ by moving $n - 2$ outside the logarithm.

$(n - 2) \ln (4) = \ln (3)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$n \ln (4) - 2 \ln (4) = \ln (3)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$n \ln (4) - 2 \ln (4) - \ln (3) = 0$

Step 5

Move all terms not containing $n$ to the right side of the equation.

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Step 5.1

Add $2 \ln (4)$ to both sides of the equation.

$n \ln (4) - \ln (3) = 2 \ln (4)$

Step 5.2

Add $\ln (3)$ to both sides of the equation.

$n \ln (4) = 2 \ln (4) + \ln (3)$

Step 6

Divide each term in $n \ln (4) = 2 \ln (4) + \ln (3)$ by $\ln (4)$ and simplify.

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Step 6.1

Divide each term in $n \ln (4) = 2 \ln (4) + \ln (3)$ by $\ln (4)$ .

$\frac{n \ln (4)}{\ln (4)} = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (3)}{\ln (4)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{n \ln (4)}{\ln (4)} = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (3)}{\ln (4)}$

Step 6.2.1.2

Divide $n$ by $1$ .

$n = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (3)}{\ln (4)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

Cancel the common factor.

$n = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (3)}{\ln (4)}$

Step 6.3.1.2

Divide $2$ by $1$ .

$n = 2 + \frac{\ln (3)}{\ln (4)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = 2 + \frac{\ln (3)}{\ln (4)}$

Decimal Form:

$n = 2.79248125 \dots$