Question

Solve each logarithmic equation. Express irrational solutions in exact form. $$ \log _{2}(5 x)=4 $$

Short answer

The solution is \( x = \frac{16}{5} \).

Step-by-step solution

Understand the Equation

The given equation is \( \log_{2}(5x) = 4 \). This is a logarithmic equation with base 2.

Convert the Logarithmic Equation to Exponential Form

Use the property of logarithms: \( \log_{b}(y) = x \) is equivalent to \( b^x = y \). Here, \( b = 2 \), \( y = 5x \), and \( x = 4 \). So, convert \( \log_{2}(5x) = 4 \) to exponential form: \( 2^4 = 5x \).

Solve the Exponential Equation

Solve for \( x \) in the equation \( 2^4 = 5x \): \[ 2^4 = 16 \] \[ 16 = 5x \] Divide both sides by 5: \[ x = \frac{16}{5} \]

Verify the Solution

To ensure the solution is correct, substitute \( x = \frac{16}{5} \) back into the original equation: \[ \log_{2}(5 \cdot \frac{16}{5}) \] Simplify: \[ \log_{2}(16) \] Since \( 2^4 = 16 \), \[ \log_{2}(16) = 4 \] which matches the original equation \( \log_{2}(5x) = 4 \). This confirms that our solution is correct.

Key concepts

logarithmic equations

Logarithmic equations involve expressions where a variable is inside a logarithm. These equations often look like \( \log_b(y) = x\). The goal is to solve for the variable, often found inside the logarithm. For instance, in the example \( \log_{2}(5x) = 4\), the variable \( x \) is inside the logarithmic expression. To solve logarithmic equations, we frequently convert them to exponential form to make them easier to handle.

Understanding the properties of logarithms is crucial. Basic properties that are helpful include:

• \( \log_b(b^x) = x \)
• \( \log_b(a \cdot c) = \log_b(a) + \log_b(c) \)
• \( a \cdot \log_b(x) = \log_b(x^a) \)

Knowing these properties helps us manipulate and solve logarithmic equations more easily.

exponential form

To solve logarithmic equations, we often need to convert them into exponential form using the property: \( \log_b(y) = x \) is equivalent to \( b^x = y\). This conversion simplifies the equation, making it easier to solve.

For the example \( \log_{2}(5x) = 4\), we use the conversion property:

• Identify \( b = 2\).
• Identify \( y = 5x\).
• Identify \( x = 4\).

Using these values, we convert the logarithmic equation into an equivalent exponential form:
\( \log_{2}(5x) = 4 \) translates to \( 2^4 = 5x \).
This transformation allows us to solve directly for the variable in the equation.

verify solution

Once we find the solution, it is essential to verify it to ensure it satisfies the original equation. Verification involves substituting the solution back into the initial logarithmic equation.

Let's review the verification process for \( x = \frac{16}{5}\).

• Substitute \( x = \frac{16}{5} \) back into \( \log_{2}(5x) = 4\).
• Simplify inside the logarithm: \( 5 \cdot \frac{16}{5} = 16 \).
• The expression becomes \( \log_{2}(16)\).
• Since \( 16 = 2^4\), we get \( \log_{2}(16) = \log_{2}(2^4) = 4 \).

Since the calculated result matches the original equation, the solution \( x = \frac{16}{5} \) is verified and correct.