Solve Solve 3 ^ y = 9 ^ - 2 Given that | 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

$$3 ^ { y } = 9 ^ { - 2 }$$

Solve for y

$y=-4$

Steps Using Definition of Logarithm

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

$$3^{y}=\frac{1}{81}$$

Take the logarithm of both sides of the equation.

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

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

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

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

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

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

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

Solve for y (complex solution)

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