Problem 36 - Algebra 2

Question

Evaluate each expression. \(\log _{12} 144\)

Step-by-Step Solution

Answer

The answer is 2.

Step 1: Understand What We Need to Find

We need to find the value of \( \log_{12} 144 \). This means we need to determine the power to which the base 12 must be raised to get 144.

Step 2: Rewrite the Logarithmic Equation

Rewrite the equation \( \log_{12} 144 = x \) in exponential form as \( 12^x = 144 \). This means we are looking for \( x \) such that when 12 is raised to the power \( x \), the result is 144.

Step 3: Express 144 Using Base 12

Try to express 144 as a power of 12. Start by factorizing 144: 144 = 12 x 12 = 12².So, we can express 144 as \( 12^2 \).

Step 4: Solve for x

Now that we have \( 12^x = 12^2 \), we can equate the exponents since the bases are the same. Therefore, \( x = 2 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are closely related to exponential functions. If you have ever wondered about the natural connection between exponents and logarithms, understanding a logarithmic function is an excellent starting point. A logarithmic function is expressed in the form \( \log_b a = x \), where \( b \) is the base, \( a \) is the argument, and \( x \) is the exponent. In simpler words, it tells us what power we need to raise the base \( b \) to in order to get the number \( a \).
Important properties of logarithms include:

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \)
• Quotient Rule: \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \)
• Power Rule: \( \log_b (m^n) = n \cdot \log_b m \)
• Change of Base: \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) can be any positive number

These properties help in simplifying complex logarithmic expressions, making calculations more manageable. This is particularly useful when solving logarithmic equations like \( \log_{12} 144 \). By leveraging these rules, we can often find simpler expressions or values without much direct computation.

Exponential Equations

When we talk about exponential equations, we are referring to equations where the variable appears in the exponent. These equations can often be rewritten as logarithmic equations to simplify the process of finding the variable. For the example \( 12^x = 144 \), we have an exponential equation because the variable \( x \) is the exponent. The goal is typically to isolate \( x \) to find its value.

Here are key strategies for solving exponential equations:

• If the bases on both sides of the equation are the same, then you can simply set the exponents equal to each other, as we did in the solution when we found that \( 12^x = 12^2 \) led to \( x = 2 \).
• If the bases are not immediately clear or different, consider rewriting one or both sides using a common base or applying logarithms.
• Use logarithmic properties to handle more complex equations, especially when common bases aren't obvious or when working with non-integer solutions.

Exponential equations are a key component in a variety of real-world applications, from calculating compound interest to modeling population growth.

Logarithmic Equations

Logarithmic equations involve variables inside the logarithm and are solved by applying the definition of logarithms or by utilizing logarithmic properties. Given an equation like \( \log_{12} 144 = x \), we can use the property of logarithms that allows us to convert it into an exponential equation like \( 12^x = 144 \), making it sometimes easier to solve.

These are the steps to typically solve logarithmic equations:

• Convert the logarithmic equation into its equivalent exponential form. This often simplifies the problem, converting an equation with a logarithm into one that can be solved with straightforward algebraic methods.
• If your equation has multiple logarithms, use logarithmic identities to combine them into a single logarithm or to eliminate logarithmic terms.
• Once simplified, solve the resulting equation using algebraic techniques, checking the solution back in the original equation to ensure it doesn't incorporate undefined logarithms, such as the log of a negative number.

Understanding logarithmic equations can make tackling various mathematical and scientific problems easier, as they allow us to transition between multiplicative and additive interpretations of numbers.