Problem 72 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( x = \frac{3}{2} \)

Step 1: Rewrite the Logarithm in Exponential Form

The equation given is \( \log_{4}{8} = x \). To convert the logarithmic equation to exponential form, remember that \( \log_{b}{a} = c \) is equivalent to \( b^{c} = a \). Hence, rewrite the equation as \( 4^{x} = 8 \).

Step 2: Express Both Sides with the Same Base

Recognize that both 4 and 8 can be written as powers of 2. This way, you express 4 as \( 2^{2} \) and 8 as \( 2^{3} \). Substitute these back into the equation to get \( (2^{2})^{x} = 2^{3} \).

Step 3: Simplify the Equation Using Exponent Laws

Use the exponent rule \((a^m)^n = a^{m \cdot n}\) to simplify \( (2^2)^x \), which becomes \( 2^{2x} \). Now the equation is \( 2^{2x} = 2^{3} \).

Step 4: Equate the Exponents

Since the bases are the same, set the exponents equal to each other: \( 2x = 3 \).

Step 5: Solve for x

Divide both sides of the equation by 2 to solve for \( x \): \( x = \frac{3}{2} \).

Key Concepts

Logarithms

Logarithms are a way to express exponential relationships. They are the inverse functions of exponentials. This means that the logarithm of a number is the exponent to which the base must be raised to get that number. For example, in the equation \( \log_{4} 8 = x \), we are saying that \( 4 \) raised to the power of \( x \) equals \( 8 \).
To convert a logarithmic equation into an exponential form, remember the basic formula: if \( \log_{b} a = c \), then it is also true that \( b^c = a \). This is a fundamental principle that allows you to move easily between logarithmic and exponential forms.
Logarithms help us to solve equations where the unknown variable appears as an exponent. When the bases are different or unmanageable, expressing both sides with a common base often simplifies the problem.

Exponential Equations

Exponential equations are ones in which variables appear as exponents. They can sometimes be tricky because they require transforming the equation into a more manageable form.
For example, given the logarithmic equation \( \log_{4}{8} = x \), we rewrite it as an exponential equation: \( 4^{x} = 8 \). This step makes it clear that we need to find the value of \( x \) such that when \( 4 \) is raised to the power of \( x \), it equals \( 8 \).
To handle exponential equations effectively, it often helps to express all numbers involved as powers of a common base. In this case, recognizing that both \( 4 \) and \( 8 \) can be written using the base \( 2 \) simplifies the solving process drastically.

Exponent Laws

Understanding exponent laws is crucial for simplifying exponential expressions. Key laws include:

• \( (a^m)^n = a^{m \cdot n} \), which allows you to multiply the exponents when an exponent is raised to another power.
• If two powers with the same base are equal, their exponents must be equal. In mathematical terms, if \( a^m = a^n \), then \( m = n \).

In our solution, we used the exponent law \( (2^2)^x = 2^{2x} \) to simplify the expression \( (2^2)^x = 2^{3} \) to \( 2^{2x} = 2^{3} \). Once you have the bases equal, the exponents must be set equal to solve for the unknown variable.
Exponent laws are powerful tools for simplifying and solving equations that involve powers, whether they show up directly like \( 4^x = 8 \), or indirectly through transformations and algebraic manipulation.