Solve ((1)/(3))^(x)=(1)/(81) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$${ \left( \frac{ 1 }{ 3 } \right) }^{ x } = \frac{ 1 }{ 81 }$$

Solve for x

$x=4$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{3}\right)^{x}=\frac{1}{81}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{x})=\log(\frac{1}{81})$$

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

$$x\log(\frac{1}{3})=\log(\frac{1}{81})$$

Divide both sides by $\log(\frac{1}{3})$.

$$x=\frac{\log(\frac{1}{81})}{\log(\frac{1}{3})}$$

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

$$x=\log_{\frac{1}{3}}\left(\frac{1}{81}\right)$$

Solve for x (complex solution)

$x=-\frac{2\pi n_{1}i}{\ln(3)}+4$
$n_{1}\in \mathrm{Z}$