Problem 72 - Intermediate Algebra

Question

Solve for \(x .\) See Example 3. $$ \log _{4} 8=x $$

Step-by-Step Solution

Answer

The value of \(x\) is \(\frac{3}{2}\).

Step 1: Understand the Problem

We need to solve the equation \(\log_{4} 8 = x\). This means we are looking for the power \(x\) that 4 must be raised to, to result in 8.

Step 2: Change of Base Formula

We use the change of base formula to convert the logarithm to a fraction. The change of base formula is: \[\log_{b} a = \frac{\log_{c} a}{\log_{c} b} \]For this problem, we will use base 2, so \[\log_{4} 8 = \frac{\log_{2} 8}{\log_{2} 4}\]

Step 3: Simplify the Numerator

Calculate \(\log_{2} 8\). Since 8 is \(2^3\), we have \[\log_{2} 8 = 3\]

Step 4: Simplify the Denominator

Calculate \(\log_{2} 4\). Since 4 is \(2^2\), we have \[\log_{2} 4 = 2\]

Step 5: Solve for x

Insert the simplified logs back into the fraction: \[\log_{4} 8 = \frac{3}{2}\]Thus, the solution is \(x = \frac{3}{2}\).

Key Concepts

Change of Base Formula

The change of base formula is a very useful tool when working with logarithms, especially when you don't have access to a calculator with specific base capabilities. Essentially, this formula allows you to rewrite a logarithm in terms of any base you wish. It states that a logarithm of a number with a certain base can be expressed using a different base, typically more convenient for calculations. The formula is presented as:

• \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \)

In practical terms, if you want to calculate \( \log_{4} 8 \) but find it difficult to do so directly, you can use the change of base formula to convert it to a base you find easier, like base 2 or 10. This flexibility is particularly helpful when solving equations or when you need a specific base for consistency in calculations. For our problem, converting \( \log_{4} 8 \) to base 2 simplifies our job because both 4 and 8 are powers of 2.

Logarithmic Equations

Logarithmic equations involve variables within the log expression itself. In many ways, solving these equations can be compared to solving algebraic equations, but they have their own unique set of rules. The main task often involves isolating the variable by using properties of logarithms and exponents, or altering the equation using the change of base formula.

• Learning how to manipulate logarithms helps in simplifying and solving such equations.
• Recognizing patterns like \( \log_{b} b = 1 \) or \( \log_{b} 1 = 0 \) can be quite handy in solutions.

In the example \( \log_{4} 8 = x \), we want to express the relation between 4, 8, and the unknown \( x \) through exponentiation. By converting the logarithmic equation into a base we're comfortable with and using known logarithmic values, we can find a straightforward answer for \( x \). Engaging in practice problems like this is a great way to become more familiar with solving different logarithmic equations.

Exponents

Exponents are fundamentally associated with logarithms as they represent the power a number (the base) is raised to, creating another number. In the context of logarithms, finding the logarithm of a number essentially asks "to what power must the base be raised to achieve this number?" If you have \( \log_{b} a = x \), it translates to the exponential form \( b^x = a \).

• This understanding is key when converting between logarithmic and exponential forms.
• Identifying the common powers in expressions simplifies the computation significantly.

For instance, in our case, knowing \( 8 = 2^3 \) and \( 4 = 2^2 \) allows us to express the original equation in forms that make calculations far easier (using smaller numbers or simpler powers). This type of manipulation is crucial, especially as you advance in mathematics, where identifying such relationships supports more complex problem-solving.