Problem 8 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. For \(5^{x}=2,\) the power rule for logarithms provides a way of moving the variable \(x\) from its position as an ______ to a position as a factor. b. If the power rule for logarithms is used on the left side of the equation \(\log 5^{x}=2,\) the resulting equation is \(\square\) \(\log 5=2\)

Step-by-Step Solution

Answer

a. exponent b. x \cdot log 5 = 2

Step 1: Identify the Exponent

In the expression \(5^x = 2\), identify \(x\) as the exponent that is being moved by the power rule for logarithms.

Step 2: Apply the Power Rule

The power rule of logarithms states that \(\log_b(a^c) = c \cdot \log_b(a)\). Here, apply it to \(\log 5^x\) to rewrite it as \(x \cdot \log 5\).

Step 3: Complete the Equation

In the equation \(\log 5^x = 2\), using the power rule the resulting equation becomes \(x \cdot \log 5 = 2\).

Key Concepts

Power Rule

The power rule for logarithms is a handy tool in mathematics, especially when dealing with expressions and equations involving exponents. It allows us to simplify and manipulate complex logarithmic expressions by bringing down exponents as factors in front of the logarithm. This is particularly useful when you want to solve equations where the variable is an exponent.

Think of the power rule as a friend who makes your math homework easier. The rule is stated as follows: for any positive number \( a \), \( b \) (where \( b eq 1 \)) and any real number \( c \), the expression \( \log_b(a^c) \) can be simplified to \( c \cdot \log_b(a) \).

• "\( c \)" is the exponent that was attached to \( a \).
• The "\( \log_b \)" represents the logarithm with base \( b \).
• Using the power rule helps to "untangle" \( c \) from the exponential spot, moving it to a more manageable place as a multiplier.

This process is like solving a puzzle, where you reposition the pieces—here, the exponent—in a way that reveals the solution more clearly.

Exponent

Exponents might seem like a small detail, but they hold significant power (pun intended) in mathematics. An exponent is a tiny number that sits slightly raised to the right of your base number, like \( x \) in \( 5^x \). This little number tells you how many times to multiply the base by itself.

For example:

• In \( 5^3 \), the exponent is 3, meaning you multiply 5 by itself three times: \( 5 \times 5 \times 5 = 125 \).
• The exponent "dictates" the repetition of multiplication, making it a concise way to express large numbers.

In logarithmic equations, the challenge often lies in the exponent being a variable. That's where the power rule comes in. It moves the exponent out of its initial position, making it less mysterious and easier to work with. An exponent may seem small, but its impact is magnified through mathematical expressions.

Logarithmic Equations

Logarithmic equations are equations that involve the logarithm of a variable or a number. They can look slightly intimidating at first but are fascinating once you get to understand them.

The key to solving logarithmic equations often lies in converting them to a more straightforward form. This is where rules like the power rule often come into play. Instead of grappling with complicated forms, you can utilize the properties of logarithms to simplify them:

• Consider an equation like \( \log 5^x = 2 \). Without any context, it might seem daunting.
• Applying the power rule changes the equation to \( x \cdot \log 5 = 2 \). This is a much cleaner and more approachable equation.
• Now, you can proceed to solve for \( x \), making it a more straightforward algebraic equation.

Ultimately, mastering logarithmic equations means mastering the art of manipulation using logarithm laws and rules, allowing you to tackle even the trickiest problems with confidence.