Question

question
evaluate:
log_{243} 3
answer attempt 1 out of 2

Explanation:

Step1: Recall the definition of logarithm

Let \( \log_{a}b = x \), which means \( a^{x}=b \). For \( \log_{243}3 \), we let \( \log_{243}3 = x \), so by the definition of logarithm, we have \( 243^{x}=3 \).

Step2: Express 243 as a power of 3

We know that \( 243 = 3^{5} \), so substitute \( 243 \) with \( 3^{5} \) in the equation \( 243^{x}=3 \), we get \( (3^{5})^{x}=3 \).

Step3: Simplify the left - hand side

According to the power - of - a - power rule \( (a^{m})^{n}=a^{mn} \), \( (3^{5})^{x}=3^{5x} \). So our equation becomes \( 3^{5x}=3^{1} \).

Step4: Solve for x

Since if \( a^{m}=a^{n} \) (where \( a>0,a
eq1 \)), then \( m = n \). Here \( a = 3 \), \( m = 5x \), \( n = 1 \), so we have \( 5x=1 \). Solving for \( x \), we get \( x=\frac{1}{5} \).

Answer:

\(\frac{1}{5}\)