Problem 7 - Algebra 2

Question

Given \(\log _{2} 7 \approx 2.8074\) and \(\log _{5} 8 \approx 1.2920\) to approximate the value of each expression. \(\log _{5} 64\)

Step-by-Step Solution

Answer

\(\log_{5} 64 \approx 2.5840\)

Step 1: Express 64 as a power of 8

Recognize that 64 can be expressed as a power of 8: \(64 = 8^{2}\). This will allow us to use the property of logarithms involving exponents.

Step 2: Use the Change of Base Formula

The Change of Base Formula states \(\log_{b} a = \frac{\log_{k} a}{\log_{k} b}\) for any positive \(a, b, k\). Use the known values \(\log_{5} 8\) and \(\log_{5} 64 = \log_{5} (8^2)\) to express \(\log_{5} 64 = 2 \cdot \log_{5} 8\).

Step 3: Substitute Known Values

Substitute the approximate value of \(\log_{5} 8 \approx 1.2920\) from the given information into the expression for \(\log_{5} 64\). Calculate \(\log_{5} 64 = 2 \cdot 1.2920 = 2.5840\).

Step 4: Conclude the Calculation

Having substituted and calculated, conclude that \(\log_{5} 64 \) is approximately \(2.5840\).

Key Concepts

Change of Base Formula

The change of base formula is a very handy tool when dealing with logarithms, especially when you need to work with different bases. This formula allows us to convert logarithms of any base to a more familiar one, commonly base 10 or base e (natural logarithms), which are easily computed with calculators.

The formula is: \[\log_{b} a = \frac{\log_{k} a}{\log_{k} b}\]In this formula, "b" is the base we are converting from, "a" is the number we are taking the logarithm of, and "k" is the new base of the logarithm we are converting to.

• The numerator, \(\log_{k} a\), is the logarithm of "a" with the new base "k".
• The denominator, \(\log_{k} b\), is the logarithm of the old base "b" with the new base "k".

Using this formula allows us to solve complex logarithmic equations more easily by changing them into computations involving more manageable bases.

Properties of Logarithms

Understanding the fundamental properties of logarithms can simplify operations and calculations involving logarithms. These properties help break down complicated logarithmic expressions into simpler terms.

Some key properties include:

• Product Property: \(\log_{b} (xy) = \log_{b} x + \log_{b} y\). This property states that the logarithm of a product is the sum of the logarithms.
• Quotient Property: \(\log_{b} \left(\frac{x}{y}\right) = \log_{b} x - \log_{b} y\). Here, the logarithm of a quotient is the difference of the logarithms.
• Power Property: \(\log_{b} (x^c) = c \cdot \log_{b} x\). This allows you to move exponents in a logarithmic argument to the front as a multiplier.

These properties are like special rules that govern how logarithms behave. They are particularly useful when solving logarithmic equations or when you're dealing with powers, roots, and products/divisions of numbers.

Exponents

Exponents are a shorthand way to indicate repeated multiplication of the same number by itself. In the context of logarithms, exponents play a crucial role as they are often involved in simplifying logarithmic expressions.

For example, writing a number like 64 as an exponent, such as \(8^2\), can often reveal relationships between numbers that simplify logarithmic calculations. Exponential expressions make it easier to see how certain numbers can factor into problems involving logarithms.

• An expression like \(a^b\) indicates "a multiplied by itself b times."
• Logarithms are essentially “undoing” the process of applying an exponent (i.e., figuring out what power you need to raise a number to get another number).

When dealing with logarithms, interpreting and manipulating exponential forms can simplify what might seem like a complex problem, especially when using properties like the Power Property of Logarithms.