Basic Math Examples

$3^{3 \sqrt{{(4 - a)}^{2}} + 10} = 19$

Step 1

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

$\ln (3^{3 \sqrt{{(4 - a)}^{2}} + 10}) = \ln (19)$

Step 2

Expand the left side.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(4 - a)}^{2}}$ as ${(4 - a)}^{\frac{2}{2}}$ .

$\ln (3^{3 {(4 - a)}^{\frac{2}{2}} + 10}) = \ln (19)$

Step 2.2

Divide $2$ by $2$ .

$\ln (3^{3 {(4 - a)}^{1} + 10}) = \ln (19)$

Step 2.3

Expand $\ln (3^{3 {(4 - a)}^{1} + 10})$ by moving $3 {(4 - a)}^{1} + 10$ outside the logarithm.

$(3 {(4 - a)}^{1} + 10) \ln (3) = \ln (19)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(3 {(4 - a)}^{1} + 10) \ln (3)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify.

$(3 (4 - a) + 10) \ln (3) = \ln (19)$

Step 3.1.1.2

Apply the distributive property.

$(3 \cdot 4 + 3 (- a) + 10) \ln (3) = \ln (19)$

Step 3.1.1.3

Multiply $3$ by $4$ .

$(12 + 3 (- a) + 10) \ln (3) = \ln (19)$

Step 3.1.1.4

Multiply $- 1$ by $3$ .

$(12 - 3 a + 10) \ln (3) = \ln (19)$

Step 3.1.2

Simplify by multiplying through.

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Step 3.1.2.1

Add $12$ and $10$ .

$(- 3 a + 22) \ln (3) = \ln (19)$

Step 3.1.2.2

Apply the distributive property.

$- 3 a \ln (3) + 22 \ln (3) = \ln (19)$

Step 4

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

$- 3 a \ln (3) + 22 \ln (3) - \ln (19) = 0$

Step 5

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

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Step 5.1

Subtract $22 \ln (3)$ from both sides of the equation.

$- 3 a \ln (3) - \ln (19) = - 22 \ln (3)$

Step 5.2

Add $\ln (19)$ to both sides of the equation.

$- 3 a \ln (3) = - 22 \ln (3) + \ln (19)$

Step 6

Divide each term in $- 3 a \ln (3) = - 22 \ln (3) + \ln (19)$ by $- 3 \ln (3)$ and simplify.

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Step 6.1

Divide each term in $- 3 a \ln (3) = - 22 \ln (3) + \ln (19)$ by $- 3 \ln (3)$ .

$\frac{- 3 a \ln (3)}{- 3 \ln (3)} = \frac{- 22 \ln (3)}{- 3 \ln (3)} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 3 a \ln (3)}{- 3 \ln (3)} = \frac{- 22 \ln (3)}{- 3 \ln (3)} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{a \ln (3)}{\ln (3)} = \frac{- 22 \ln (3)}{- 3 \ln (3)} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.2.2

Cancel the common factor of $\ln (3)$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{a \ln (3)}{\ln (3)} = \frac{- 22 \ln (3)}{- 3 \ln (3)} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 22 \ln (3)}{- 3 \ln (3)} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Cancel the common factor of $\ln (3)$ .

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Step 6.3.1.1.1

Cancel the common factor.

$a = \frac{- 22 \ln (3)}{- 3 \ln (3)} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.3.1.1.2

Rewrite the expression.

$a = \frac{- 22}{- 3} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.3.1.2

Dividing two negative values results in a positive value.

$a = \frac{22}{3} + \frac{\ln (19)}{- 3 \ln (3)}$

Step 6.3.1.3

Move the negative in front of the fraction.

$a = \frac{22}{3} - \frac{\ln (19)}{3 \ln (3)}$

Step 7

Exclude the solutions that do not make $3^{3 \sqrt{{(4 - a)}^{2}} + 10} = 19$ true.

$No solution$