Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Approximate the logarithm using the properties of logarithms, given the values \(\log _{b} 2 \approx 0.3562\) \(\log _{b} 3 \approx 0.5646,\) and \(\log _{b} 5 \approx 0.8271 .\) Round your result to four decimal places.$$\log _{b} 8$$.

Step-by-Step Solution

Answer

The approximation of \(\log_b(8)\) to four decimal places is \(1.0686\).

Step 1: Express the unknown logarithm in terms of known logarithms

Using the property of logarithms that \(\log_b(a^n) = n \cdot \log_b(a)\), we can express \(\log_b(8)\) as \(\log_b(2^3)\) which then equals \(3 \cdot \log_b(2)\).

Step 2: Substitute the given value

Substitute the given value \(\log_b(2) = 0.3562\) into your previous result, which gives: \(3 \cdot \log_b(2) = 3 \cdot 0.3562\).

Step 3: Calculate the result

Calculate \(3 \cdot 0.3562\) which equals \(1.0686\). Remember to round your answer to four decimal places.

Key Concepts

Logarithm Approximation

Logarithm approximation is often used when you only have certain known logarithm values and you need to compute the logarithm of another number. To do this efficiently, we make use of logarithm properties and express the unknown logarithm in terms of these known values.

For example, if we need to approximate \( \log_b(8) \) and we know the value of \( \log_b(2) \), we can use the logarithmic property which tells us that \( \log_b(a^n) = n \cdot \log_b(a) \). Thus, since 8 is equal to \( 2^3 \), we can rewrite the equation as \( \log_b(8) = \log_b(2^3) \) and then simplify it to \( 3 \cdot \log_b(2) \). With this approach, approximation becomes a simple calculation step once we have the value of \( \log_b(2) \).

• Use known logarithm values.
• Apply logarithmic properties.
• Express unknown in terms of known values.

Logarithms Calculation

Calculating logarithms involves using specific mathematical properties to determine the power to which a base number is raised to produce a given number.

In practical scenarios like the given exercise, you substitute the known values into your expression to find the result. For instance, if you are provided that \( \log _{b} 2 \approx 0.3562 \), which means that the base \( b \) must be raised to the power of approximately 0.3562 to equal 2, the task might be to find \( \log _{b} 8 \). By substituting the given approximation into the expression, it simplifies the calculation process.

Here are some steps you can follow:

• Decipher the base and known logarithm values.
• Express complex logarithms using known values.
• Substitute and compute accurately.
• Always remember to round your final answer correctly.

Exponent Rules

The properties of exponents provide a bridge between multiplication and exponentiation, which are crucial for working with logarithms effectively.

When calculating logarithms such as \( \log_b(a^n) \), the exponent rules tell us that this is equivalent to \( n \cdot \log_b(a) \). This shows us that exponents can be treated as coefficients outside the logarithm.

Understanding this rule is fundamental when you need to simplify expressions for approximation or calculation, as it allows you to break down powers into manageable parts that align with known values.

For instance, using the rule that \( a^n \) inside a logarithm can be pulled out as a multiplier, we can solve \( \log_b(8) = \log_b(2^3) = 3 \cdot \log_b(2) \), making it straightforward to substitute and find results.

• Recognize powers and simplify with logarithms.
• Use rules to convert complex expressions.
• Treat powers as coefficients for ease of calculation.