Problem 94 - Elementary and Intermediate Algebra

Question

Assume that \(\log 4 \approx 0.6021, \log 7 \approx 0.8451,\) and \(\log 9 \approx 0.9542 .\) Use these values to evaluate each logarithm. \(\log _{b} 49\)

Step-by-Step Solution

Answer

The evaluation of \(\log_b 49\) depends on the base \(b\); if known, use \(\log_b 49 = \frac{1.6902}{\log b}\).

Step 1: Recall the Change of Base Formula

To solve for \(\log_b 49\), we need to use the change of base formula, which states that \(\log_b 49 = \frac{\log 49}{\log b}\).

Step 2: Express 49 in terms of 7

Recognize that 49 can be written as \(7^2\). Therefore, \(\log 49 = \log(7^2)\).

Step 3: Use the Power Rule for Logarithms

According to the power rule, \(\log(7^2) = 2 \cdot \log 7\). Substitute the given values to get \(2 \cdot 0.8451 = 1.6902\).

Step 4: Find the Value of \(\log_b 49\)

Substitute the value of \(\log 49\) obtained from Step 3 into the change of base formula: \(\log_b 49 = \frac{1.6902}{\log b}\). Since the exercise does not provide a value for \(\log b\), the solution for \(\log_b 49\) depends on the base \(b\).

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in mathematics that allows us to evaluate logarithms with different bases by converting them into a ratio of logarithms with a common base, typically 10 or e. This is especially useful when we only have logarithms available in base 10 (common logarithms) or base e (natural logarithms).

To use the change of base formula, the expression \(\log_b(a)\) can be rewritten as:

• \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\)

where \(c\) can be any base, but common choices are base 10 or e.
In practical situations, this formula lets you compute logarithms with non-standard bases using a calculator that only calculates base 10 or base e logarithms. For example, in the given exercise, you find \(\log_b 49 = \frac{1.6902}{\log b}\), using the computed \(\log 49\) from the step-by-step solution.

This is crucial for problems involving logarithms with bases other than 10 or e, especially when needing exact or approximate values.

Power Rule for Logarithms

The power rule for logarithms simplifies logarithmic expressions that involve exponents. It states that if you are taking a logarithm of an exponent, you can "bring down" the exponent as a multiplier of the logarithm.

Mathematically, the power rule can be expressed as:

• \(\log_b(x^n) = n \cdot \log_b(x)\)

This rule is handy because it transforms complex expressions into simpler ones, making them easier to evaluate.
In the exercise, you were given that 49 equals \(7^2\). Applying the power rule, the expression \(\log(7^2)\) became \(2 \cdot \log 7\).

By knowing \(\log 7 = 0.8451\), you substituted this back to get \(2 \cdot 0.8451 = 1.6902\). This is how using the power rule reduces the problem to simpler calculations, making it manageable even without a calculator that handles complex logarithms directly.

Evaluating Logarithms

Evaluating logarithms involves finding the numeric value of a logarithmic expression. Often, you need to employ formulas or known logarithm values, such as the power rule or change of base formula, to simplify and compute these expressions.
For the exercise, you were given approximations for several logarithms like \(\log 4, \log 7,\) and \(\log 9\). These helped in evaluating \(\log(7^2)\), transforming it into a simpler form using the power rule and approximated value of \(\log 7\).

To evaluate a logarithm step by step:

• Look for ways to express the number in simpler terms (e.g., as a power of a known base).
• Use properties and rules of logarithms, like the power rule.
• If necessary, apply the change of base formula to make calculations easier.

Remember, evaluating logarithms effectively means using the given information and logarithmic properties to simplify and calculate values, often leading to exact answers or tangible approximations when precise calculation isn't feasible.