Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{5} 12 $$

Step-by-Step Solution

Answer

\( \log_{5} 12 \approx 1.5798 \)

Step 1: Recognize the Formula

We need to calculate \( \log_{5} 12 \) using the formula \( \log_{a} x = \frac{\ln x}{\ln a} \). This formula allows us to express a logarithm with any base in terms of natural logarithms.

Step 2: Substitute Values into the Formula

Substitute \( x = 12 \) and \( a = 5 \) into the formula. This gives us: \[ \log_{5} 12 = \frac{\ln 12}{\ln 5} \].

Step 3: Calculate Natural Logarithms

Calculate the natural logarithms (\( \ln \)) of 12 and 5. Using a calculator, we find: \( \ln 12 \approx 2.4849 \) and \( \ln 5 \approx 1.6094 \).

Step 4: Compute the Quotient

Divide the natural logarithm of 12 by the natural logarithm of 5: \[ \frac{2.4849}{1.6094} \approx 1.5798 \].

Step 5: Verify the Solution

Make sure that the calculation is precise by checking that the computed value of \( \log_{5} 12 \) rounds to approximately the same number across various calculations or tools.

Key Concepts

Natural Logarithms

Natural logarithms, denoted by \( \ln \), are a special kind of logarithm where the base \( e \) is used. The number \( e \) (approximately 2.718) is a fundamental constant in mathematics, much like \( \pi \). It's derived from the concept of continuous growth and is invaluable in calculus, complex systems, and many other fields.
Natural logarithms help simplify complex exponential equations. They can convert multiplication processes into simple addition, which is particularly useful for solving exponential growth problems.

• For example, \( \ln(e^x) = x \) because raising \( e \) to the power of \( x \) exactly reverses the logarithm of \( x \).
• This makes \( \ln \) extremely useful in various scientific and engineering applications.

Using \( \ln \) is especially practical in digital computations, showing how logs translate multiplication into addition.

Change of Base Formula

The change of base formula is a handy mathematical tool that allows us to calculate logarithms of any base using natural logarithms. The formula is expressed as:
\[ \log_{a} x = \frac{\ln x}{\ln a} \]
Here,\( a \) is the base of the logarithm, and \( x \) is the number being evaluated. This beautiful formula helps us convert a log of one base to another using natural log values.
Understand this process:

• By using \( \ln \), you eliminate the need for multiple logarithmic tables.
• This is practical for computations where standard calculators have built-in natural logarithms and base 10 logarithms only.
• Example: To find \( \log_{5} 12 \) conventionally can be cumbersome. Instead, use \( \frac{\ln 12}{\ln 5} \) for a swift calculation.

Mastering this formula is essential for handling complex algebraic transformations elegantly.

Calculator Usage in Mathematics

Calculators are indispensable in modern mathematics, whether dealing with basic arithmetic or complex functions like logarithms. Since built-in functions such as \( \ln \) and \( \log \) are readily available, it simplifies many computations significantly.
When using a calculator to calculate logarithms:

• Identify the required function (\( \ln \) for natural logs or \( \log \) for base 10 logs).
• Input the value you wish to determine the logarithm for.
• Directly retrieve accurate decimal equivalent values – for example, \( \ln 12 \approx 2.4849 \).

Most scientific calculators enable you to utilize the change of base formula effectively, mainly relying on \( \ln \). This integration allows us to determine non-standard logarithmic calculations without extra tables or computational aids.