Question

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562\), \(\log _{b} 3 \approx 0.5646\), and \(\log _{b} 5 \approx 0.8271$$\log _{b} 40\)

Short answer

The approximate value of \(log_b40\) is 1.8957.

Step-by-step solution

Understand the problem and properties of logarithms

We know that the basic rules of logarithms include: \[the \text{ sum rule, \(log_b(MN) = log_bM + log_bN\)}\], \[the \text{ difference rule, \(log_b(M/N) = log_bM - log_bN\)}\], and \[the \text{ power rule, \(log_b(M^p) = p*log_bM\)}\]. Given, we have \(\log _{b} 2 \approx 0.3562, \log _{b} 3 \approx 0.5646, \text{ and } \log _{b} 5 \approx 0.8271\). We need to find \(log_b40\) using the properties of logarithms.

Express the number 40 in terms of 2,3 and 5

The number 40 can be expressed as a product of 2,3 and 5. We have \[40 = 2^3 * 5\]. We now have the product that we can substitute into the logarithmic expression \(log_b40\).

Use the properties of logarithms to evaluate \(log_b40\)

Substitute 40 with \(2^3 * 5\) in \(log_b40\) to yield \[log_b(2^3*5) = log_b(8*5).\] Use the sum rule of logarithm to split the above expression into the sum of two logs: \[log_b(8*5) = log_b8 + log_b5.\] Again, use the power rule to simplify \(log_b8\) to \(3log_b2\). Final expression becomes \[3*log_b2 + log_b5.\]

Substitute given values into the expression

Plug in the given values for \(log_b2\) and \(log_b5\) into the expression \[3 * log_b2 + log_b5 = 3 * 0.3562 + 0.8271.\]

Perform the calculation

Multiply and add the values to get the final result for \(log_b40\). We get \[1.0686 + 0.8271 = 1.8957.\]

Key concepts

Properties of Logarithms

Logarithms have specific properties that make them a powerful tool for simplifying complex equations.
These properties allow us to manipulate and combine logarithmic expressions in ways that can simplify calculations and problem-solving.
Some of the essential properties of logarithms include:

• Sum Rule: If you have a product inside the logarithm, you can split it into two separate logarithms with addition, expressed as: \(\log_b(M \cdot N) = \log_bM + \log_bN\).
• Difference Rule: When dealing with a quotient inside a logarithm, you can separate it into two logarithms with subtraction, noted as: \(\log_b(\frac{M}{N}) = \log_bM - \log_bN\).
• Power Rule: When a logarithm contains an exponent, you can pull the exponent in front of the logarithm, formulated as: \(\log_b(M^p) = p \cdot \log_bM\).

These rules are powerful because they transform difficult multiplicative and divisive relationships into simpler additive and subtractive ones.
Understanding these properties is key to solving logarithmic problems efficiently.

Logarithm Rules

Logarithmic rules are fundamental guidelines that help us understand how logarithms behave in mathematical operations.
These rules are derived from the properties of logarithms and are crucial in handling expressions involving logarithmic functions.
Let's delve into a few key rules:

• Change of Base Formula: This rule helps in computing logarithms with different bases: \(\log_bM = \frac{\log_kM}{\log_kb}\), where \(k\) is any positive number.
• Inverse Rule: This states that a logarithm of a power with the same base is equal to the exponent itself: \(b^{\log_bM} = M\). This indicates the inverse nature of logarithms and exponents.
• Log of One Rule: It asserts that any log of 1 in any base will always be zero: \(\log_b1 = 0\).

These rules assist in converting complex logarithmic expressions into easier computations.
They enable converting any logarithmic expression to another base if needed, and help understand the foundational nature of logarithms.

Logarithmic Approximation

Logarithmic approximation is a technique used to estimate logarithmic values when exact values are not necessary or when they are difficult to calculate.
This practice is particularly helpful when dealing with logarithms in bases other than 10 or \(e\), which are commonly found in standard tables or calculators.
Here's an approach to logarithmic approximation:

• Using Known Values: If you know the logarithmic values for specific bases, you can approximate the log of other numbers by breaking them down into smaller factors. For instance, if \(\log_b 2\), \(\log_b 3\), and \(\log_b 5\) are known, you can approximate \(\log_b 40\) as described in the exercise by expressing 40 as \(2^3 \cdot 5\).
• Properties of Logarithms: Utilize properties such as the sum, difference, and power rules to break down and approximate the values.

By leveraging known logarithmic values and the properties of logarithms, one can efficiently approximate the logarithm of complex expressions.
This technique is valuable when precise accuracy is not the primary concern, and it paves the way for easier calculations in both academic and practical applications.