Solve (2)^(x)\times(2)^(2)\times5+((2)^(x))/((2)^(2))=(13)/(16) | 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

$${ 2 }^{ x } \times { 2 }^{ 2 } \times 5+ \frac{ { 2 }^{ x } }{ { 2 }^{ 2 } } = \frac{ 13 }{ 16 }$$

Solve for x

$x=\log_{2}\left(\frac{13}{324}\right)\approx -4.639410285$

Steps Using Definition of Logarithm

Calculate $2$ to the power of $2$ and get $4$.

$$2^{x}\times 4\times 5+\frac{2^{x}}{2^{2}}=\frac{13}{16}$$

Multiply $4$ and $5$ to get $20$.

$$2^{x}\times 20+\frac{2^{x}}{2^{2}}=\frac{13}{16}$$

Calculate $2$ to the power of $2$ and get $4$.

$$2^{x}\times 20+\frac{2^{x}}{4}=\frac{13}{16}$$

Combine $2^{x}\times 20$ and $\frac{2^{x}}{4}$ to get $\frac{81}{4}\times 2^{x}$.

$$\frac{81}{4}\times 2^{x}=\frac{13}{16}$$

Multiply both sides by $\frac{4}{81}$, the reciprocal of $\frac{81}{4}$.

$$2^{x}=\frac{13}{16}\times \frac{4}{81}$$

Multiply $\frac{13}{16}$ and $\frac{4}{81}$ to get $\frac{13}{324}$.

$$2^{x}=\frac{13}{324}$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(\frac{13}{324})$$

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

$$x\log(2)=\log(\frac{13}{324})$$

Divide both sides by $\log(2)$.

$$x=\frac{\log(\frac{13}{324})}{\log(2)}$$

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

$$x=\log_{2}\left(\frac{13}{324}\right)$$

Solve for x (complex solution)

$x=\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(\frac{13}{324}\right)$
$n_{1}\in \mathrm{Z}$