Questions: Find the exact value of each logarithm without a calculator. a. log base 7 of 49 b. log base 4 of 4 c. log base 1/2 of 32

Solution Steps

To find the exact value of each logarithm, we need to determine the exponent that the base must be raised to in order to get the given number.

a. For $\log _{7} 49$, we need to find the exponent \( x \) such that \( 7^x = 49 \).

b. For $\log _{4} 4$, we need to find the exponent \( x \) such that \( 4^x = 4 \).

c. For $\log _{1 / 2} 32$, we need to find the exponent \( x \) such that \( (1/2)^x = 32 \).

To find \(\log _{7} 49\), we need to determine the exponent \( x \) such that \( 7^x = 49 \).

Since \( 49 = 7^2 \), we have: \[ 7^x = 7^2 \] Thus, \( x = 2 \).

To find \(\log _{4} 4\), we need to determine the exponent \( x \) such that \( 4^x = 4 \).

Since \( 4 = 4^1 \), we have: \[ 4^x = 4^1 \] Thus, \( x = 1 \).

To find \(\log _{1 / 2} 32\), we need to determine the exponent \( x \) such that \( \left(\frac{1}{2}\right)^x = 32 \).

Since \( 32 = 2^5 \), we can rewrite the equation as: \[ \left(\frac{1}{2}\right)^x = 2^5 \] This implies: \[ 2^{-x} = 2^5 \] Thus, \( -x = 5 \) and \( x = -5 \).

Final Answer