Question

evaluate the following logarithms without a calculator (no decimal answers):
$7^{\log_{7}(13)} = \square$
$e^{\ln(6)} = \square$
question help: \\( \boldsymbol{\text{video}} \\)

Explanation:

Step1: Recall the inverse property of logarithms and exponents

For any positive number \( a \) (where \( a
eq 1 \)) and any real number \( x \), the property \( a^{\log_a(x)} = x \) holds. Also, for the natural logarithm, since \( \ln(x)=\log_e(x) \), the property \( e^{\ln(x)}=x \) holds.

Step2: Evaluate \( 7^{\log_7(13)} \)

Using the property \( a^{\log_a(x)} = x \) with \( a = 7 \) and \( x = 13 \), we get \( 7^{\log_7(13)}=13 \).

Step3: Evaluate \( e^{\ln(6)} \)

Using the property \( e^{\ln(x)} = x \) with \( x = 6 \), we get \( e^{\ln(6)} = 6 \).

Answer:

For \( 7^{\log_7(13)} \), the answer is \( 13 \). For \( e^{\ln(6)} \), the answer is \( 6 \).