Trigonometry Examples

$4^{- r + 3} = 1$

Step 1

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

$\ln (4^{- r + 3}) = \ln (1)$

Step 2

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

$(- r + 3) \ln (4) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$- r \ln (4) + 3 \ln (4) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

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

$- r \ln (4) + 3 \ln (4) = 0$

Step 5

Subtract $3 \ln (4)$ from both sides of the equation.

$- r \ln (4) = - 3 \ln (4)$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Divide $r$ by $1$ .

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

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.

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

Step 6.3.1.2

Rewrite the expression.

$r = \frac{- 3}{- 1}$

Step 6.3.1.3

Move the negative one from the denominator of $\frac{- 3}{- 1}$ .

$r = - 1 \cdot - 3$

Step 6.3.2

Simplify the expression.

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Step 6.3.2.1

Rewrite $- 1 \cdot - 3$ as $- - 3$ .

$r = - - 3$

Step 6.3.2.2

Multiply $- 1$ by $- 3$ .

$r = 3$