Questions: For log base 3 of 5, (a) Estimate the value of the logarithm between two consecutive integers.

Transcript text: For $\log _{3} 5$ ， (a) Estimate the value of the logarithm between two consecutive integers.

Solution

Solution Steps

Step 1: Estimation between two consecutive integers

For $\log_{3} 5$, we find the smallest integer $n$ such that $3^{n} > 5$. The logarithm will be between 1 and 2.

Step 2: Approximation using the change-of-base formula

Using the change-of-base formula, $\log_{3} 5 = \frac{\log_{k} 5}{\log_{k} 3}$, where $k$ is a common base. The approximation to 4 decimal places is 1.465.

Step 3: Checking the result with the exponential form

Converting the logarithmic statement into its exponential form, $3^{c} = 5$, where $c$ is the approximation. Calculating $3^{c}$ gives 5, which should be close to 5 if the approximation holds true.

Final Answer:

The estimated range for $\log_{3} 5$ is between 1 and 2, and its approximation to 4 decimal places is 1.465. Checking the result with the exponential form gives 5, validating our approximation.

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