Question

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{\sqrt{2}} 7\)

Short answer

5.615

Step-by-step solution

Understand the Change-of-Base Formula

The Change-of-Base Formula allows you to evaluate logarithms with any base using common logarithms (base 10) or natural logarithms (base e). For a logarithm \( \log_{a} b \ \), the formula is: \[ \log_{a} b = \frac{\log b}{\log a} \ \].

Identify the Base and Argument

In the given logarithm, \( \log_{\sqrt{2}} 7 \ \), the base \( \sqrt{2} \ \) and the argument \( 7 \ \) have been identified.

Apply the Change-of-Base Formula

Rewrite the logarithm using the Change-of-Base Formula: \[ \log_{\sqrt{2}} 7 = \frac{\log 7}{\log \sqrt{2}} \ \].

Evaluate Using a Calculator

Using a calculator, find \log 7 \ and \log \sqrt{2}: \[ \log 7 \approx 0.845098 \ \] and \[ \log \sqrt{2} \approx 0.150515 \ \].

Calculate the Result

Divide the results from Step 4: \[ \frac{0.845098}{0.150515} \ \approx 5.615 \ \].

Round the Answer

Round the answer to three decimal places: \[ 5.615 \ \].

Key concepts

Logarithms

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

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.

Calculator Usage

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.