Problem 42 - Intermediate Algebra

Question

For Problems \(41-50\), solve each equation. $$ \log _{2} x=5 $$

Step-by-Step Solution

Answer

The solution to \( \log_{2} x = 5 \) is \( x = 32 \).

Step 1: Understand the equation

The given equation is \( \log_{2} x = 5 \). This means that the logarithm of \( x \) with base 2 equals 5. We need to find the value of \( x \).

Step 2: Convert from logarithmic to exponential form

Recall that the expression \( \log_{b} a = c \) is equivalent to saying \( b^{c} = a \). Apply this principle: \( \log_{2} x = 5 \) becomes \( 2^{5} = x \).

Step 3: Calculate the power

Compute \( 2^{5} \): \[ 2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32 \].

Step 4: Write the solution

Thus, the value of \( x \) is 32, so \( x = 32 \).

Key Concepts

Understanding Logarithms

Logarithms are a way to solve equations involving exponents. If you have an equation like \( \log_{b} x = y \), it means you're trying to find what power you need to raise the base \( b \) to in order to get \( x \). In simpler terms, it's asking: "What power of \( b \) equals \( x \)?"

• The number \( b \) is the base of the logarithm.
• The number \( x \) is what the logarithm equals when \( b \) is raised to a certain power.
• The result, \( y \), is the power that \( b \) needs.

This makes logarithms incredibly useful for solving equations where the unknown is an exponent. By converting logarithmic equations to their exponential form, they become much easier to tackle.

Exponential Form Conversion

To solve logarithmic equations, converting them to exponential form can be very helpful. The general rule is if you have \( \log_{b} a = c \), it can be rewritten as \( b^{c} = a \).

• The base \( b \) is the same in both forms.
• The result \( a \) is what's achieved by raising \( b \) to the power of \( c \).
• This conversion helps in visualizing the problem more easily and sets up the equation for solving.

In the given problem \( \log_{2} x = 5 \), it converts to \( 2^{5} = x \). This transformation lets us directly compute the value of \( x \) by calculating powers, which often feels more straightforward.

Solving Equations Efficiently

Solving equations requires understanding the path from the problem to the answer. After converting logarithmic equations to exponential form, the main task is to solve for the unknown. By computing the powers, you're essentially completing the equation.

• For \( \log_{2} x = 5 \), we convert it to \( 2^{5} = x \).
• Next, calculate \( 2^5 \). Multiplying, \( 2 \times 2 \times 2 \times 2 \times 2 \) gives us \( 32 \).
• This shows that the value of \( x \) is \( 32 \).

Once you've solved for \( x \), you can check your work by substituting back. It confirms that you're correct if it holds true in the original equation. This method ensures clarity and confidence in solving similar problems.