1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

Solve 3 त | Equatix

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Question

$$( \sqrt { 3 } ) ^ { 5 } \times 9 ^ { 2 } = 3 ^ { n } \times 3 \sqrt { 3 }$$

Solve for n

$n=5$

Steps Using Definition of Logarithm

Calculate $9$ to the power of $2$ and get $81$.

$$\left(\sqrt{3}\right)^{5}\times 81=3\times 3^{n}\sqrt{3}$$

Swap sides so that all variable terms are on the left hand side.

$$3\times 3^{n}\sqrt{3}=\left(\sqrt{3}\right)^{5}\times 81$$

Subtract $\left(\sqrt{3}\right)^{5}\times 81$ from both sides.

$$3\times 3^{n}\sqrt{3}-\left(\sqrt{3}\right)^{5}\times 81=0$$

Multiply $-1$ and $81$ to get $-81$.

$$3\times 3^{n}\sqrt{3}-81\left(\sqrt{3}\right)^{5}=0$$

Use the rules of exponents and logarithms to solve the equation.

$$3\sqrt{3}\times 3^{n}-81\left(\sqrt{3}\right)^{5}=0$$

Subtract $-81\left(\sqrt{3}\right)^{5}$ from both sides of the equation.

$$3\sqrt{3}\times 3^{n}=729\sqrt{3}$$

Divide both sides by $3\sqrt{3}$.

$$3^{n}=243$$

Take the logarithm of both sides of the equation.

$$\log(3^{n})=\log(243)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$n\log(3)=\log(243)$$

Divide both sides by $\log(3)$.

$$n=\frac{\log(243)}{\log(3)}$$

By the change-of-base formula $\frac{\log(a)}{\log(b)}=\log_{b}\left(a\right)$.

$$n=\log_{3}\left(243\right)$$

Show Solution

Solve for n (complex solution)

$n=\frac{2\pi n_{1}i}{\ln(3)}+5$
$n_{1}\in \mathrm{Z}$