Question

If y is a negative number such that \({2^{{y^2}{{\log }_3}5}} = 5{ ^{lo{g_2}3}}\), then y equals
1. - log₂(1/5)
2. log₂(1/5)
3. -log₂(1/3)
4. log₂(1/3)

Answer 1

Correct Answer - Option 3 : -log₂(1/3)

Calculation:

\({2^{{y^2}{{\log }_3}5}} = 5{ ^{lo{g_2}3}}\)

a^x = N

x = log_aN

According to the properties

⇒ y²log₃5 = log₂5^log₂3

⇒ y2log35 = log₂3 × log₂5 (by using log_aM^x = xlog_aM)

⇒ y² = (log23 × log25)/log35

⇒ y² = {(log_e3/log_e2) × (log_e5/loge2)/(loge​5/log_e3)}

⇒ y² = (loge3/loge2)²

⇒ y = (+ or –) loge3/loge2

If y is negative

⇒ y = – log₂(1/3)

∴ The value of y is – log2(1/3).