Solve (256) ^ (x/4) = 1/64 | 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

$$(256)^{\frac{x}{4}}=\frac{1}{64}$$

Solve for x

$x=-3$

Steps Using Definition of Logarithm

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

$$256^{\frac{1}{4}x}=\frac{1}{64}$$

Take the logarithm of both sides of the equation.

$$\log(256^{\frac{1}{4}x})=\log(\frac{1}{64})$$

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

$$\frac{1}{4}x\log(256)=\log(\frac{1}{64})$$

Divide both sides by $\log(256)$.

$$\frac{1}{4}x=\frac{\log(\frac{1}{64})}{\log(256)}$$

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

$$\frac{1}{4}x=\log_{256}\left(\frac{1}{64}\right)$$

Multiply both sides by $4$.

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

Solve for x (complex solution)

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