What does logb(x^b) simplify to?

To understand why the expression ( \log_b(x^b) ) simplifies to ( x ), we can apply the properties of logarithms. One of the key properties states that the logarithm of a number raised to an exponent can be rewritten using the exponent as a multiplier:

[

\log_b(a^n) = n \cdot \log_b(a)

]

In this case, we can let ( a = x ) and ( n = b ). Thus, we can rewrite the expression:

[

\log_b(x^b) = b \cdot \log_b(x)

]

Next, we utilize another logarithmic property: ( \log_b(b) = 1 ). Since ( x ) is what we are taking the log of, if we assume ( x = b ), we simplify further:

[

\log_b(b) = 1 \quad \implies \quad \log_b(x^b) = b \cdot 1 = b

]

However, when the log base ( b ) is used for ( x^b ), we can simplify this directly back to:

[

\log_b(b^1) = 1