Explain why b^x = e^xlnb.

Verified step by step guidance

1. Recall the definition of the natural exponential function, where e is the base of the natural logarithm and ln is the natural logarithm function.
2. Recognize that any exponential function can be expressed in terms of the natural exponential function using the change of base formula.
3. Use the property of logarithms that states ln(a^b) = b * ln(a) to rewrite b^x as e raised to the power of x times ln(b).
4. Express b^x as e^(ln(b^x)), which simplifies to e^(x * ln(b)) using the logarithmic identity mentioned.
5. Conclude that b^x = e^(x * ln(b)), which confirms that b^x can be expressed as e^x * ln(b).