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Logarithms are mathematical expressions used to denote the power to which a base number must be raised to obtain a given number. For instance, in the expression \(\text{log}_{10}100 \), the base is 10, and the logarithm tells us that 10 must be raised to the power of 2 to equal 100. Logarithms have properties that make them useful in various mathematical applications, including solving exponential equations and dealing with large numbers.
Some essential properties of logarithms are:

• \(\text{log}_{a}(xy) = \text{log}_{a}x + \text{log}_{a}y \)
• \(\text{log}_{a}(x/y) = \text{log}_{a}x - \text{log}_{a}y \)
• \(\text{log}_{a}(x^{y}) = y \text{log}_{a}x \)

Mastering these properties will help you manipulate and understand logarithmic functions efficiently.

Base conversion in logarithms involves changing the base of a given logarithm to a more convenient base—usually 10 (common logarithms) or e (natural logarithms). This is achieved using the Change-of-Base Formula:
\[ \text{\text{log}_{a}b} = \frac{\text{log}b}{\text{log}a} \] This formula allows you to convert any logarithm to one with a base that your calculator or mathematical software supports. For example, to evaluate \(\text{\text{log}_{\text{\text{sqrt{2}}}}7} \), you rewrite it as:
\[ \text{\text{log}_{\text{\text{sqrt{2}}}}7 = \frac{\text{log}7}{\text{log}{\text{sqrt{2}}}} \]
This step makes the calculation straightforward, as most calculators can easily compute logs with the base 10 or e.

Using a calculator to evaluate logarithms involves a few crucial steps to ensure accuracy:

• Input the argument of the logarithm first. For example, to find \(\text{log}7 \), type '7' followed by the log function on your calculator.
• Repeat the same for the base part. For instance, for \(\text{log}{\text{sqrt{2}}} \), first calculate \(\text{sqrt{2}} \) and then apply the log function to it.
• Divide the result of the argument log by the result of the base log as indicated by the Change-of-Base Formula.
• Round your final answer to the required decimal places—usually this means rounding to three decimal places.

For the example given, \(\text{\text{log}_{\text{sqrt{2}}}}7 \), you would enter it as:
\[ \frac{\text{\text{log}7}}{\text{\text{log}{\text{sqrt{2}}}}} = \frac{0.845098}{0.150515} \]
This gives a result of approximately 5.615 when rounded to three decimal places.