Question

use common logarithms or natural logarithms and a calculator to evaluate the express try again. rewrite the given expression using the change of base property and then evaluate. recall that the change of base property introducing common logarithms is \\(\log_b m = \frac{\log m}{\log b}\\) and introducing natural logarithms is \\(\log_b m = \frac{\ln m}{\ln b}\\).

Explanation:

Step1: Identify the logarithm

Suppose we have a logarithm \(\log_{b}M\) (the original problem's specific \(\log_{b}M\) isn't fully shown, but the process is general). Let's assume we need to evaluate \(\log_{b}M\) (e.g., if \(b = 5\), \(M=12\) as a sample case).

Step2: Apply change - of - base (common log)

Using the change - of - base formula for common logarithms \(\log_{b}M=\frac{\log M}{\log b}\). For our sample case \(\log_{5}12=\frac{\log 12}{\log 5}\).

Step3: Calculate with calculator

Using a calculator, \(\log 12\approx1.0792\), \(\log 5\approx0.6990\). Then \(\frac{\log 12}{\log 5}=\frac{1.0792}{0.6990}\approx1.544\).
(If we use natural logarithms: \(\log_{b}M = \frac{\ln M}{\ln b}\), for \(\log_{5}12=\frac{\ln 12}{\ln 5}\), \(\ln 12\approx2.4849\), \(\ln 5\approx1.6094\), \(\frac{2.4849}{1.6094}\approx1.544\))

Since the original problem's specific logarithm expression is not fully provided (the "express" part in "evaluate the express" is cut off), but the general process is as above. Let's assume the actual problem is to evaluate a logarithm like \(\log_{a}c\) (where \(a\) and \(c\) are numbers). The key steps are:

1. Identify the base \(b\) and the argument \(M\) of the logarithm \(\log_{b}M\).
2. Apply the change - of - base formula (either \(\frac{\log M}{\log b}\) or \(\frac{\ln M}{\ln b}\)).
3. Use a calculator to find the values of the logarithms in the numerator and denominator and then divide.

For example, if the problem was \(\log_{3}8\):

• Using common logs: \(\log_{3}8=\frac{\log 8}{\log 3}\), \(\log 8\approx0.9031\), \(\log 3\approx0.4771\), \(\frac{0.9031}{0.4771}\approx1.893\)
• Using natural logs: \(\log_{3}8 = \frac{\ln 8}{\ln 3}\), \(\ln 8\approx2.0794\), \(\ln 3\approx1.0986\), \(\frac{2.0794}{1.0986}\approx1.893\)

Answer:

(The answer depends on the specific logarithm expression. For example, if the expression is \(\log_{5}12\), the answer is approximately \(1.544\); if it is \(\log_{3}8\), the answer is approximately \(1.893\). The general process is to apply the change - of - base formula and then use a calculator to compute the quotient.)