Question

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{1 / 2} 15\)

Short answer

-3.907

Step-by-step solution

Identify the Change-of-Base Formula

The Change-of-Base Formula is: \[ \log_b{a} = \frac{ \log_c{a} }{ \log_c{b} } \] We can choose any base (often base 10 or base e) to convert the logarithm.

Apply the Change-of-Base Formula

To evaluate \log_{1/2}{15}\ using the change-of-base formula with base 10, rewrite as follows: \[ \log_{1/2}{15} = \frac{ \log_{10}{15} }{ \log_{10}{1/2} } \]

Calculate the Numerator

Use a calculator to find the logarithm of 15 with base 10: \[ \log_{10}{15} \approx 1.176 \]

Calculate the Denominator

Use a calculator to find the logarithm of 1/2 with base 10: \[ \log_{10}{1/2} \approx -0.301 \]

Divide the Values

Now divide the results obtained: \[ \log_{1/2}{15} = \frac{1.176}{-0.301} \approx -3.907 \]

Key concepts

Logarithms

Logarithms are mathematical tools that help us determine the power to which a number (base) must be raised to obtain another number. They are the inverse operations of exponentiation. For example, if we have \(2^3 = 8\), then the logarithm base 2 of 8 is written as \( \text{log}_2 8 \) and equals 3.
They allow us to simplify complex calculations, especially those involving multiplication, division, and exponentiation. They are widely used in various fields such as science, engineering, and financial modeling.
Logarithms come with properties that make them quite useful, like:

• Product Rule: \( \text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y) \)
• Quotient Rule: \( \text{log}_b(x/y) = \text{log}_b(x) - \text{log}_b(y) \)
• Power Rule: \( \text{log}_b(x^r) = r \text{log}_b(x) \)

Understanding and working with logarithms can greatly simplify solving exponential equations.

Base Conversion

Base conversion is a useful technique for evaluating logarithms when the given base is not common, like 10 (common logarithm) or e (natural logarithm). This is where the Change-of-Base Formula comes into play. The formula is expressed as:
\( \text{log}_b{a} = \frac{ \text{log}_c{a} }{ \text{log}_c{b} } \)
Here, \( b \) and \( c \) are the bases, and \( a \) is the argument we want the logarithm of. Often, we select base 10 or base e because these are easy to compute using calculators.
For example, to convert \( \text{log}_{1/2}{15} \) to base 10, the formula alters the expression to:

• \( \text{log}_{1/2}{15} = \frac{ \text{log}_{10}{15} }{ \text{log}_{10}{1/2} } \)

Now both the numerator and denominator can be calculated using a calculator, thus simplifying the problem.

Calculator Use

Using a calculator is essential when dealing with logarithms, especially for non-standard bases that require conversion. Most scientific calculators have dedicated logarithm functions for base 10 (log) and base e (ln).
To evaluate \( \text{log}_{10}{15} \), simply input 15 and press the 'log' button:

• \( \text{log}_{10}{15} \) will yield approximately 1.176

Similarly, to find \( \text{log}_{10}{1/2} \), input 0.5 and press the 'log' button:

• \( \text{log}_{10}{1/2} \) will yield approximately -0.301

After obtaining these results, you can then perform the division:
\( \frac{1.176}{-0.301} \) which approximately equals -3.907.
Ensure accuracy by double-checking these values, and remember to round your answers to the desired decimal places as instructed by the problem. Calculators make it significantly easier to work with complex expressions and non-standard logarithmic bases.