Solve 1/16 ^ (3x - 1) = 32 | 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

$$\frac{1}{16}^{3x-1}=32$$

Solve for x

$x=-\frac{1}{12}\approx -0.083333333$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{16}\right)^{3x-1}=32$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{16}\right)^{3x-1})=\log(32)$$

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

$$\left(3x-1\right)\log(\frac{1}{16})=\log(32)$$

Divide both sides by $\log(\frac{1}{16})$.

$$3x-1=\frac{\log(32)}{\log(\frac{1}{16})}$$

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

$$3x-1=\log_{\frac{1}{16}}\left(32\right)$$

Add $1$ to both sides of the equation.

$$3x=-\frac{5}{4}-\left(-1\right)$$

Divide both sides by $3$.

$$x=-\frac{\frac{1}{4}}{3}$$

Solve for x (complex solution)

$x=-\frac{i\pi n_{1}}{6\ln(2)}-\frac{1}{12}$
$n_{1}\in \mathrm{Z}$