Problem 16 - College Algebra

Question

For the following exercises, rewrite each equation in logarithmic form. $$ 4^{x}=y $$

Step-by-Step Solution

Answer

Logarithmic form: \( x = \log_{4}(y) \)

Step 1: Understand the Exponential Form

The given equation is in exponential form: \( 4^x = y \). In this expression, 4 is the base, \( x \) is the exponent, and \( y \) is the result.

Step 2: Recall the Logarithmic Form

The logarithmic form of an equation is \( \log_{base}(result) = exponent \). You want to find \( \log_{4}(y) = x \).

Step 3: Convert to Logarithmic Form

Using the formula from Step 2, rewrite the exponential equation \( 4^x = y \) as \( x = \log_{4}(y) \). This is the equivalent logarithmic form.

Key Concepts

Exponential Form

Exponential form is a way of expressing numbers using a base and an exponent. It takes the format of \( a^b = c \). Here, \( a \) is the "base," \( b \) is the "exponent," and \( c \) is the "result" of the exponentiation. If you see an expression like \( 4^x = y \), it's saying that 4 is being multiplied by itself \( x \) times to get \( y \). This form is common in mathematics, particularly in fields dealing with growth, decay, and repeated multiplication.
Understanding exponential form is essential for converting to other mathematical notations, such as logarithmic form. By identifying the base, exponent, and result, you create a clear path for rearranging the equation into another format that might simplify solving it or apply to a different context.

Base of Logarithm

The "base" of a logarithm refers to the number that is raised to a certain power to obtain another number. In the logarithmic form, \( \log_a(b) = c \), \( a \) is the base. It represents the repeated factor you're using. The base is equivalent to the base in the corresponding exponential form \( a^c = b \). It signifies what potential value has been used to obtain the result through multiplication.

• For example, in \( \log_4(y) = x \), the base is 4. This tells us that 4 is repeatedly multiplied \( x \) times to result in \( y \).

It's crucial to accurately identify the base as it sets the "foundation" for what you're "building" with your exponents. This understanding enables easier conversion between exponential and logarithmic equations.

Converting Equations to Logarithms

Converting an equation from exponential to logarithmic form involves rearranging the components in a way that represents the same relationship differently. This process is important as it helps in solving equations that might be complex in exponential form.

• Start by identifying the components of the exponential form \( a^b = c \): the base \( a \), the exponent \( b \), and the result \( c \).
• Use the logarithmic form \( \log_a(c) = b \) to express the same relationship. Here, the base \( a \) remains the same, \( c \) is the result that now becomes part of the logarithm, and \( b \) is the exponent converted to the result of the logarithm.
• For example, from \( 4^x = y \), converting gives \( x = \log_4(y) \). The base 4 remains constant, \( y \) moves inside the logarithm, and \( x \) is the equation's solution.

This conversion isn't merely a rearrangement; it can make some problems easier to understand or solve, particularly in practical applications like computing compound interest or solving for time in exponential growth models.