Problem

Use common logarithms or natural logarithms and a calculator to evaluate the expression. \[ \log _{18} 16 \] Use the change-of-base property to rewrite the given expression in terms of natural logarithms or common logarithms. $\log _{18} 16=$ (Do not evaluate.)

Solution

Step 1 :Given the expression \(\log _{18} 16\)

Step 2 :We can use the change-of-base property to rewrite the given expression in terms of natural logarithms or common logarithms. The change-of-base formula is \(\log_b a = \frac{\log_k a}{\log_k b}\), where k can be any positive number except 1.

Step 3 :Using this formula, we can rewrite the expression \(\log _{18} 16\) as follows:

Step 4 :Using natural logarithms (base e): \(\log _{18} 16 = \frac{\ln 16}{\ln 18}\)

Step 5 :Using common logarithms (base 10): \(\log _{18} 16 = \frac{\log 16}{\log 18}\)

Step 6 :\(\boxed{\log _{18} 16 = \frac{\ln 16}{\ln 18}}\) or \(\boxed{\log _{18} 16 = \frac{\log 16}{\log 18}}\)

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Source: https://solvelyapp.com/problems/8127/

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