Problem 1 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{6} 36=2 $$

Step-by-Step Solution

Answer

The exponential equation is \(6^2 = 36\).

Step 1: Understand the Logarithmic Statement

The logarithmic equation \( \log_{b}(a) = c \) translates to "\(c\) is the power to which the base \(b\) must be raised to produce the number \(a\)." In this exercise, we have \( \log_{6}(36) = 2 \), meaning 2 is the power to which 6 must be raised to get 36.

Step 2: Translate to Exponential Form

To convert the logarithmic statement into an exponential equation, rewrite \( \log_{6}(36) = 2 \) as \( 6^2 = 36 \). This shows the base, 6, raised to the power of 2, gives 36.

Step 3: Confirm the Translation

Check if the exponential equation correctly represents the original statement. By calculating \(6^2\), we see that it equals 36, confirming \(6^2 = 36\) is the correct exponential form of the logarithmic equation.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms. A common form of a logarithmic equation is \( \log_b(a) = c \), where \(b\) is the base, \(a\) is the result, and \(c\) is the exponent indicating the power needed for the base to reach \(a\). Logarithms answer the question: "To what power must the base be raised to obtain a specific number?"
Understanding logarithms helps in solving for unknown exponents, which is crucial when dealing with exponential growth or decay. For instance, in our original exercise \( \log_{6}(36) = 2 \), the logarithm expresses that 6 must be squared to result in 36.
Logarithms have several important properties:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Property: \( \log_b(M^k) = k\log_b(M) \)

Exponential Form

The exponential form is the result of writing a logarithm expression as an exponent. In our example, converting \( \log_{6}(36) = 2 \) into exponential form shows the relationship as \( 6^2 = 36 \). This means you are raising the base, 6, to the power of 2 to arrive at 36.
Exponential equations are powerful in demonstrating how quantities grow or shrink. Recognizing and manipulating exponential forms is key to solving a wide range of problems in science and finance.
To switch from a logarithmic equation to its exponential form:

• Identify the base \(b\), the result \(a\), and the exponent \(c\).
• Rearrange into \(b^c = a\). In our example, it’s \(6^2 = 36\).
• Verify your conversion by calculating the right-hand side.

Base and Exponent

The base and exponent are two fundamental components of exponential expressions. The base is the number being multiplied, while the exponent indicates how many times the base is used in the multiplication.
In the example \(6^2 = 36\), the base is 6, and the exponent is 2, meaning you multiply 6 by itself once (i.e., \(6 \times 6\)). This multiplication results in 36.
When dealing with logarithmic or exponential equations, correctly identifying the base and exponent is crucial. They determine the structure and outcome of your calculation. Whether solving for an unknown base or exponent, retain clarity on how these components interact.

Algebra

Algebra serves as a foundation for solving both logarithmic and exponential equations. It involves manipulating symbolic equations to find unknown values.
Key skills in algebra include:

• Simplifying Equations: Break down complex equations into simpler parts for easier manipulation.
• Rearranging Equations: Logical reordering of equations to solve for a desired variable.
• Verifying Solutions: Always perform a check to confirm that your solution satisfies the original equation.

In the given logarithmic and exponential equation translation, algebra allows us to transition between forms. First, we interpret \( \log_{6}(36) = 2 \) to find its exponential counterpart \(6^2 = 36\). By practicing algebra, you enhance your ability to reason and solve more complex problems effectively.