Problem 125 - Precalculus

Question

How can the logarithmic equation \(\log _{b} x=y\) be solved for \(x\) using the properties of exponents?

Step-by-Step Solution

Answer

x = b^y.

Step 1: Understand the Given Equation

The given logarithmic equation is \( \log_{b} x = y \). This means that \( b \) raised to the power of \( y \) equals \( x \).

Step 2: Use the Definition of Logarithms

According to the definition of logarithms, the equation \( \log_{b} x = y \) can be rewritten in its exponential form. The base \( b \), when raised to the exponent \( y \), results in the number \( x \).

Step 3: Express in Exponential Form

To express the logarithmic equation in exponential form, set \( b^y = x \). This directly follows from the fact that \( \log_{b} x = y \) implies \( b^y = x \).

Step 4: Solve for x

Now that we have the equation in exponential form \( b^y = x \), it clearly shows \( x \) is equal to \( b \) raised to the power of \( y \).

Step 5: Conclude the Solution

The value of \( x \) is determined by \( x = b^y \). This is the solution obtained by converting the logarithmic equation to its exponential form using properties of exponents.

Key Concepts

Properties of Exponents

Exponents are fundamental in mathematics, especially when dealing with equations that involve powers. Understanding how to manipulate and apply them is essential for solving logarithmic equations.
Consider an expression like \( a^m \), where \( a \) is the base and \( m \) is the exponent. The exponent indicates how many times to multiply the base by itself.
Here are some key properties of exponents that are frequently used:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
• Zero Exponent: \( a^0 = 1 \) for any non-zero \( a \).
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

These properties allow you to simplify expressions and solve equations. For example, if you encounter \( b^y = x \), knowing these properties lets you manipulate \( b \) and \( y \) to extract \( x \).

Definition of Logarithms

A logarithm answers the question: "To what exponent must we raise a base to obtain a certain number?" It’s a way to reverse the process of exponentiation and relate multiplication to addition.
The notation \( \log_b x = y \) means that \( b^y = x \). Here:\

• \( b \) is the base.
• \( y \) is the exponent or logarithm.
• \( x \) is the result of raising \( b \) to the power of \( y \).

Logarithms transform multiplicative relationships into additive relationships, making complex calculations simpler. This is particularly useful in fields like engineering and computer science, where large datasets are common.
The definition clearly shows how they relate to exponentiation, which is why translating a logarithmic equation into exponential form is straightforward.

Exponential Form

Rewriting logarithmic equations in exponential form is a crucial solving strategy. By understanding that \( \log_b x = y \) implies \( b^y = x \), you convert a complex logarithmic equation into a simpler exponential one.
In the exponential form \( b^y = x \), the problem becomes finding \( x \) if \( b \) and \( y \) are known—a classic problem in algebra.
Let's consider an example:

• Suppose \( \log_2 8 = 3 \). In exponential form, this is \( 2^3 = 8 \), clearly showing that \( 3 \) is the power by which \( 2 \) must be raised to yield \( 8 \).
• This shows direct evidence of how exponentiation works in practice, reinforcing understanding and giving a concrete method to find results.

This conversion allows better insight into the structure of the problem, making it more manageable and intuitive to solve.