Solve for k: 2^k = k32

Mlinganyo uliotolewa ni:

$$2^k = k^{32}$$

$$(2^k)^{\frac{1}{32}} = (k^{32})^{\frac{1}{32}}$$ $$2^{\frac{k}{32}} = k$$

Acha $k = 2^x$. Badilisha hii kwenye mlinganyo kutoka Step 2.

$$2^{\frac{2^x}{32}} = 2^x$$

Kwa kuwa besi (base) ni sawa (2), basi exponents (nguvu) lazima ziwe sawa:

$$\frac{2^x}{32} = x$$

Tunaweza kutafuta suluhisho kwa kujaribu namba kamili (integers) au kwa kuchambua grafu za $y=2^x$ na $y=32x$.

Kujaribu namba kamili:

• Ikiwa $x=1$: $2^1 = 2$, $32 \times 1 = 32$. $2 \neq 32$.
• Ikiwa $x=2$: $2^2 = 4$, $32 \times 2 = 64$. $4 \neq 64$.
• Ikiwa $x=3$: $2^3 = 8$, $32 \times 3 = 96$. $8 \neq 96$.
• Ikiwa $x=4$: $2^4 = 16$, $32 \times 4 = 128$. $16 \neq 128$.
• Ikiwa $x=5$: $2^5 = 32$, $32 \times 5 = 160$. $32 \neq 160$.
• Ikiwa $x=6$: $2^6 = 64$, $32 \times 6 = 192$. $64 \neq 192$.
• Ikiwa $x=7$: $2^7 = 128$, $32 \times 7 = 224$. $128 \neq 224$.
• Ikiwa $x=8$: $2^8 = 256$, $32 \times 8 = 256$. Hivyo, $x=8$ ni suluhisho.

Kuna suluhisho lingine lisilo la namba kamili. Tunaweza kulipata kwa kutumia Lambert W function au kwa kukadiria.
Mlinganyo $2^x = 32x$ unaweza kuandikwa kama $x = \frac{2^x}{32}$.
Ikiwa $x$ ni ndogo sana, $2^x \approx 1$. Hivyo $x \approx \frac{1}{32}$.
Hebu tuangalie $x = \frac{1}{32}$:
$2^{\frac{1}{32}} \approx 1.0219$
$32 \times \frac{1}{32} = 1$
Kwa kuwa $1.0219 \neq 1$, $x=\frac{1}{32}$ si suluhisho kamili, lakini ni karibu. Suluhisho halisi ni takriban $x \approx 0.03158$.

Kumbuka $k = 2^x$.

Suluhisho la kwanza: $x_1 = 8$

$$k_1 = 2^{x_1} = 2^8 = 256$$

Suluhisho la pili: $x_2 \approx 0.03158$

$$k_2 = 2^{x_2} \approx 2^{0.03158} \approx 1.022$$

Kwa kawaida, katika maswali kama haya, suluhisho rahisi la namba kamili ndilo linalotarajiwa.

Jibu:
Kuna suluhisho mbili halisi kwa $k$:
$\boxed{\text{256}}$ na $\boxed{\text{1.022}}$