Problem 110 - Algebra and Trigonometry

Question

Explain how to use your calculator to find \(\log _{14} 283\)

Step-by-Step Solution

Answer

Using these steps, you'll find that \(\log_{14}283 ≈ 2.37\) when rounded to two decimal places.

Step 1: Understanding The Change of Base Formula

The Change of Base Formula is a formula that allows you to rewrite a logarithm in terms of logs written with another base. This is helpful, especially since calculators commonly don't have the option for a logarithm of base 14. The formula is as follows: \(\log_{b}a = \frac{\log_{c}a}{\log_{c}b}\), where c can be any positive number.

Step 2: Applying the Change of Base Formula

Apply the change of base formula to \(\log_{14}283\). Using the formula in step 1, it can be rewritten as \(\log_{14}283 = \frac{\log_{10}283}{\log_{10}14}\) or \(\log_{14}283 = \frac{\log_{e}283}{\log_{e}14}\) (where \(\log_{e}\) denotes the natural logarithm).

Step 3: Use Your Calculator

Enter the expression from step 2 into your calculator. If you chose the base 10 version, you should use the log button on your calculator. If you chose the natural logarithm version, use the ln button. Make sure that 283 is the numerator and 14 is the denominator.