Problem 13 - Algebra 2

Question

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 30\)

Step-by-Step Solution

Answer

\( \log_5 30 \approx 2.1133 \).

Step 1: Express 30 in terms of 2 and 3

To find the expression for 30 using the numbers with given logarithms, start by factoring 30. Notice that 30 can be expressed as the product of smaller numbers: \( 30 = 2 imes 3 imes 5 \).

Step 2: Use the properties of logarithms

Apply the logarithm property \( \log_b(mn) = \log_b m + \log_b n \). For our case, \( \log_5(30) = \log_5(2 \times 3 \times 5) \).Apply the property to break it down further into a sum:\( \log_5(30) = \log_5 2 + \log_5 3 + \log_5 5 \).

Step 3: Substitute known logarithm values

From the problem statement, we know that:\( \log_5 2 \approx 0.4307 \) and \( \log_5 3 \approx 0.6826 \).Also, by the definition of logarithms, \( \log_5 5 = 1 \) because any number's logarithm to the same base is 1.Substitute these values into the equation:\[ \log_5(30) = 0.4307 + 0.6826 + 1 \].

Step 4: Calculate the sum

Add the values together:\( 0.4307 + 0.6826 + 1 = 2.1133 \).Thus, \( \log_5 30 \approx 2.1133 \).

Key Concepts

Logarithmic Identities

Logarithmic identities provide powerful tools that can simplify and solve complex logarithmic equations.
These identities are essential in performing operations such as addition and subtraction on logarithmic expressions.

• Product Rule: The logarithm of a product is equal to the sum of the logarithms of its factors: \( \log_b(mn) = \log_b m + \log_b n \).


• Quotient Rule: The logarithm of a quotient is equal to the difference of the logarithms: \( \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \).


• Power Rule: The logarithm of a number raised to an exponent is the exponent times the logarithm of the base: \( \log_b(m^n) = n \cdot \log_b m \).

In our example, we found that \( \log_5 30 \) can be simplified by applying the product rule. By expressing 30 as a product of its factors \(2 \times 3 \times 5\), the identity allows us to break down the calculation into manageable parts. The result is a simple sum of known logarithmic values, making it easier to handle.

Change of Base Formula

The change of base formula is a useful technique that allows you to evaluate logarithms with bases that aren't directly provided by a calculator.
This formula expresses a logarithm in terms of a ratio of logarithms with a new base, usually 10 or \(e\) (natural logarithm).
The change of base formula is given by:

• \( \log_b m = \frac{\log_k m}{\log_k b} \)

This formula comes in handy when you need to compute a logarithm with an uncommon base but lack direct values on your calculator.
For instance, if you had to calculate \( \log_5 30 \) without knowing its value directly, you could use the change of base formula to rearrange it using log base 10 or natural log values. However, since this exercise already provides values, it was unnecessary. But knowing this formula ensures you can tackle any logarithmic problem with ease.

Approximation of Logarithms

Logarithms often require approximation since their exact values for certain bases and numbers are not always readily available.
Approximations help us perform mathematical operations without needing excessive computational power or tools.
To approximate logarithms efficiently:

• Use known logarithmic values, like in this example where \( \log_5 2 \approx 0.4307 \) and \( \log_5 3 \approx 0.6826 \). They're helpful for simplifying expressions.


• Combine these using logarithmic identities, as done by breaking down complex expressions like \( \log_5 30 \) into basic components. This makes calculating the overall log value straightforward.


• Remember that even approximated values provide a close enough estimation for practical uses, especially when precise tools aren't available.

By summarizing approximate values with known calculations, students can work through logarithmic tasks more easily, building a solid foundation in understanding how logs function in various mathematical contexts.