Problem 78 - Precalculus with Calculus Previews

Question

Find the numerical value of the given logarithm. $$ \log _{8} e $$

Step-by-Step Solution

Answer

The numerical value of \(\log_8 e\) is approximately 0.4816.

Step 1: Understanding the Change of Base Formula

To solve this problem, we'll use the change of base formula for logarithms, which states: \[ \log_b a = \frac{\log_k a}{\log_k b} \]where \(k\) is a new base. We'll use base 10 for simplicity.

Step 2: Applying the Change of Base Formula

Using the change of base formula, we substitute into it:\[ \log_8 e = \frac{\log_{10} e}{\log_{10} 8} \]

Step 3: Finding Logarithm Values

Calculate the values of \(\log_{10} e\) and \(\log_{10} 8\): - From a calculator, \(\log_{10} e \approx 0.4343\)- \(\log_{10} 8 = \log_{10} (2^3) = 3 \cdot \log_{10} 2 \approx 3 \cdot 0.3010 = 0.9030\)

Step 4: Calculate the Numerical Value

Substitute the values back into the formula:\[ \log_8 e = \frac{0.4343}{0.9030} \approx 0.4816 \]

Key Concepts

Logarithms

Logarithms are powerful mathematical tools used to simplify complex calculations involving exponential growth or decay. A logarithm answers the question: "To what power must a certain base be raised to produce a certain number?" For example, if we have \( a^x = b\), then \(\log_a b = x\). The base of the logarithm is crucial because it dictates the exponential rate, which is often adjusted using the change of base formula.

• Relationship with Exponents: Logarithms and exponents are inverses. For instance, \(\log_{10} 100 = 2\) means \(10^2 = 100\).
• Change of Base Formula: This is a vital method for solving logarithms, expressing a log in terms of logs of a different base.

Understanding these basics simplifies calculating the numeric value of a logarithm, like \(\log_8 e\). Once you grasp the core concepts of logarithms, interpreting solutions becomes more manageable and intuitive.

Base 10

The base 10 logarithm, often referred to as the 'common logarithm,' uses 10 as its foundation. This choice of base arises naturally due to our decimal number system. It simplifies many real-world calculations, especially those related to scientific and engineering problems.

• Symbolism: \(\log_{10} x\) is commonly expressed as \(\log x\) without the base in mathematical contexts.
• Applications: Base 10 logarithms are used to measure the magnitude of earthquakes (Richter scale), sound intensity (decibels), and in calculating pH in chemistry.

In the problem context, converting \(\log_8 e\) to base 10 offered a clearer pathway to finding its numerical value. By making calculations in this familiar base, it becomes easier to utilize technology such as calculators that directly compute \(\log_{10}e\) and \(\log_{10}8\).

Numerical Value Calculation

Calculating the numerical value of a logarithm involves applying formulas and using available computational tools. Once the change of base formula is established, inputting known numerical quantities is the next step.

• Use of Calculators: Tools can swiftly provide values for base 10 logarithms, such as \(\log_{10} e \) or \(\log_{10} 8\).
• Precision and Rounding: Calculations typically result in decimal approximations, requiring careful rounding to ensure accuracy in contexts that demand it.

After identifying \(\log_{10} e\) and \(\log_{10} 8\) using a calculator, the numerical value of \(\log_8 e\) was determined by dividing these values. The result was a rounded approximation, \(0.4816\), demonstrating the effectiveness of mathematical tools in calculating complex logarithms.