MODERATE EXERCISES-Logarithms

(2) Express without e or ln: e^{t ln 2} .

SOLUTION: First, note that we can write e^{t ln 2} as e ^{(ln 2)(t)}. Then, by the properties of exponents, e^{(ln 2)(t)} = (e ^{(ln 2)})^t. But e ^{(ln 2)} = 2 (by a property of logarithms). So:

e^{t ln 2} = e^{(ln 2)(t)} = (e ^{(ln 2)})^t =2^t.

Alternatively, you can write the exponent "t ln2" as: ln(2^t), by a property of logarithms. So then:
e^{t ln 2} = e^ln(2t) = 2^t, by another property of logarithms.