Problem 3 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Using the definition, \(x = b^y\).

Step 1: Identify the Given Equation

The given equation is a logarithmic equation: \(\log_{b} x = y\). Our task is to solve this equation for \(x\).

Step 2: Utilize the Definition of Logarithms

Use the definition of a logarithm, which states that \(\log_{b} x = y\) implies that \(b^y = x\). This is because a logarithm is the inverse operation of exponentiation.

Step 3: Express x in terms of Exponents

By using the definition, we can express \(x\) in terms of \(b\) and \(y\) as \(x = b^y\). This expression provides \(x\) directly as a power of \(b\).

Step 4: State the Final Solution for x

Hence, the solution for \(x\) in terms of \(b\) and \(y\) is \(x = b^y\). This expression states that \(x\) is the result of raising \(b\) to the power of \(y\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is key to solving logarithmic equations effectively. Exponents describe how many times to multiply a certain number, the base, by itself. For example, in the expression \(b^y\), \(b\) is the base, and \(y\) is the exponent, denoting \(b\) multiplied by itself \(y\) times.

There are several important rules involving exponents:

• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\) — Allows combining exponents when multiplying identical bases.
• Power of a Power Rule: \((a^m)^n = a^{m\times n}\) — Describes raising a power to another power.
• Power of a Product Rule: \((ab)^n = a^n \times b^n\) — Spreads the exponent across all terms in the product.
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\) — Turns a negative exponent into a reciprocal.
• Zero Exponent Rule: \(a^0 = 1\) — Any non-zero base raised to the power of zero equals one.

These exponent properties simplify complex expressions and assist in transforming logarithmic equations into exponential ones, as demonstrated in solving for \(x\) using \(b^y = x\).

Inverse Operations

Inverse operations are essential for understanding the relationship between logarithms and exponents. An inverse operation reverses the effect of the original operation. For instance, addition and subtraction are inverse operations, as they cancel each other out.

Logarithms and exponents are inverse operations of each other. If exponentiation multiplies a base \(b\) times itself, logarithms answer how many times the base needs to be multiplied to get a certain number. This inverse relationship is captured in the equation \(\log_b x = y\), which translates to \(b^y = x\).

Understanding this inverse nature helps solve equations like \(\log_b x = y\) by rewriting in exponential form \(b^y = x\). This rewrites a difficult logarithmic expression in a simpler form that is easy to work with, leading directly to the solution for \(x\).

Definition of Logarithms

The definition of logarithms is central to solving logarithmic equations. A logarithm is essentially the inverse of exponentiation. In simple terms, if you have a number \(b\) raised to the power of \(y\) that equals \(x\) (\(b^y = x\)), then the logarithm \(\log_b x\) is \(y\).

Here’s a straightforward way to think about it:

• If \(b = 2 \:and\, x = 8\), asking \(\log_2 8\) queries ‘what power of 2 gives 8?’
• The answer is 3, because \(2^3 = 8\).

Logarithms are not just theoretical; they’re practical tools used in measuring exponential growth rates and scaling in scientific calculations and computer science. Applying the logarithmic definition helps easily convert challenging equations into more manageable ones using the properties of exponents.