Problem 15 - Algebra A Combined Function

Question

Solve each equation. $$ \log _{2}(x+5)=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 11 \).

Step 1: Understand the Logarithmic Equation

The given equation is a logarithmic equation: \( \log_{2}(x + 5) = 4 \). This means that the logarithm, with base 2, of \( x + 5 \) is equal to 4.

Step 2: Convert to Exponential Form

To solve this equation, you need to convert the logarithmic equation into an exponential form. The equation \( \log_{2}(x+5) = 4 \) can be rewritten in exponential form as \( 2^4 = x + 5 \). This is because the logarithm tells us that the base raised to a power (4) equals the argument (\( x + 5 \)).

Step 3: Simplify the Exponential Equation

Calculate \( 2^4 \) to get the value of \( x + 5 \). This results in \( 2^4 = 16 \), so the equation becomes \( 16 = x + 5 \).

Step 4: Solve for x

Subtract 5 from both sides of the equation to isolate \( x \): \( 16 - 5 = x \). This results in \( x = 11 \).

Step 5: Verify the Solution

Substitute \( x = 11 \) back into the original equation to ensure it holds true: \( \log_{2}(11 + 5) = 4 \). Calculate \( 11 + 5 = 16 \) and confirm that \( \log_{2}(16) = 4 \) as \( 2^4 = 16 \). The solution is verified.

Key Concepts

Exponential Form

Converting logarithmic equations into exponential form is a crucial step in solving them. In essence, a logarithmic equation like \( \log_b(a) = c \) can be rewritten in exponential form as \( b^c = a \). This conversion leverages the fundamental relationship between logarithms and exponents.

For example, consider the equation \( \log_2(x + 5) = 4 \). The base here is 2, the argument is \( x + 5 \), and the logarithmic result is 4. By converting it to an exponential form, you are essentially asking, "What power must 2 be raised to in order to get \( x + 5 \)?" The answer is \( x + 5 = 2^4 \).

Now, 2 raised to the power of 4 is 16, simplifying our equation to \( x + 5 = 16 \). The conversion from logarithmic to exponential form is a powerful tool because it transforms an abstract logarithmic relationship into a more concrete arithmetic problem. This makes it much easier to solve.

Solving Logarithmic Equations

Solving logarithmic equations involves a few systematic steps. First, the equation needs to be identified as a logarithmic one, such as \( \log_{2}(x+5)=4 \). Recognizing the structure helps in applying the correct method.

After identifying the logarithmic equation, the next step is typically to convert the expression into exponential form, as previously discussed. This transformation simplifies the problem into a straightforward arithmetic calculation. In our example, after converting, we find that \( x + 5 = 16 \).

The subsequent steps involve solving the arithmetic equation for the variable in question. Here, we subtract 5 from both sides resulting in \( x=11 \). This process of isolation and calculation ultimately reveals the value of the unknown that satisfies the original logarithmic equation. To ensure the answer is correct, it's always good practice to substitute back the solution into the original equation to verify its validity.

Properties of Logarithms

The properties of logarithms simplify expressions and make it easier to solve equations. Understanding these properties is key to efficiently handling logarithmic equations. Here are some important properties:

• Product Property: \( \log_b(MN) = \log_bM + \log_bN \)
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_bM - \log_bN \)
• Power Property: \( \log_b(M^k) = k\cdot\log_bM \)
• Change of Base Formula: \( \log_bM = \frac{\log_cM}{\log_cb} \)

These properties allow you to manipulate logarithmic expressions to make them easier to solve. For example, when multiple logarithms are added together, you can express them as a single logarithm of a product. Similarly, subtracting logarithms can be rewritten as a single logarithm of a quotient.

Using these properties can break down complex expressions into more manageable forms, setting the stage for conversion into exponential form or further simplification. These rules make logarithmic operations more intuitive and are essential in mastering logarithmic equations.