A simple question that I do not understand why this statement is “T”

Question

It says EVERY positive a so can I just take a = 10 ?

And also it says EVERY real b so can I take b = 1 ?

However when I insert these numbers the equation is not satisfied obviously. So I thought this question is “F” but the real answer is “T”

What am I doing wrong? I do not understand what is the thing that I miss.

Thank you so much for your help.

Question:



a^b = e^{b.(log a)} , for every positive a and every real b.

TRUE or FALSE?

Answer 1

It is TRUE if you use the natural log, i.e. ln(a), and correctly apply the rules for products in exponents. Start with the expression on the right, and see that it gives the expression on the left.

e^{b ln(a)} = [e^ln(a)]^b = a^b

_________________ Addition to original answer ________________________

Consider a=10 and b=1.

a^b = 10¹ = 10

e^{b ln(a)} = [e^ln(a)]^b = [e^ln(10)]¹

By definition of natural logarithm,

e^ln(10) = 10

So

e^{b ln(a)} = 10¹ = 10

They give the same value.