Questions: Most calculators can find logarithms with base and base e. To find logarithms with different bases, we use the Select. To find log base y of 15, we write the following. (Round your answers to three decimal places.) log base 9 of 16 = log() / log() = Do we get the same answer if we perform the calculation in part (a) using ln in place of log? Yes, the result is the same. No, the result is not the same.

Solution Steps

To find the logarithm of a number with a base that is not directly supported by most calculators, we can use the change of base formula. The change of base formula states that \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), where \(c\) is a base that the calculator can handle, such as 10 or \(e\). For part (a), we will use this formula to calculate \(\log_9(16)\). For part (b), we will check if using the natural logarithm (base \(e\)) instead of the common logarithm (base 10) gives the same result.

To find \(\log_9(16)\), we use the change of base formula:
\[ \log_9(16) = \frac{\log_{10}(16)}{\log_{10}(9)} \]

Calculate \(\log_{10}(16)\) and \(\log_{10}(9)\) using a calculator:
\[ \log_{10}(16) \approx 1.2041 \]
\[ \log_{10}(9) \approx 0.9542 \]

Substitute the values into the formula:
\[ \log_9(16) = \frac{1.2041}{0.9542} \approx 1.2619 \]

Round the result to three decimal places:
\[ \log_9(16) \approx 1.262 \]

Verify the result using natural logarithms:
\[ \log_9(16) = \frac{\ln(16)}{\ln(9)} \]
Calculate \(\ln(16)\) and \(\ln(9)\):
\[ \ln(16) \approx 2.7726 \]
\[ \ln(9) \approx 2.1972 \]
Substitute the values:
\[ \log_9(16) = \frac{2.7726}{2.1972} \approx 1.2619 \]
Round the result:
\[ \log_9(16) \approx 1.262 \]

Both methods yield the same rounded result:
\[ \log_9(16) \approx 1.262 \]
Thus, the answer is the same whether using \(\log\) or \(\ln\).

Final Answer