Questions: Evaluate the expression. log16 17 ≈ (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.)

Evaluate the expression.
log16 17 ≈
(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.)

Transcript text: Evaluate the expression. \[ \log _{16} 17 \approx \square \] (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Apply the Change of Base Formula

To evaluate \(\log_{16} 17\), we use the change of base formula: \[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \] where \(b = 16\) and \(a = 17\). We can use the natural logarithm (base \(e\)) for this calculation: \[ \log_{16} 17 = \frac{\ln 17}{\ln 16} \]

Step 2: Calculate the Natural Logarithms

Calculate \(\ln 17\) and \(\ln 16\): \[ \ln 17 \approx 2.8332 \] \[ \ln 16 \approx 2.7726 \]

Step 3: Divide the Logarithms

Divide the natural logarithm of 17 by the natural logarithm of 16: \[ \log_{16} 17 = \frac{2.8332}{2.7726} \approx 1.0219 \]