Problem 23 - Precalculus

Question

Use the properties of logarithms to write the expression as a single logarithm. $$ \log _{5}(x)-3 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{5}(125x) \).

Step 1: Understand the Expression

The problem is asking us to combine the logarithmic term and the numeric term into a single logarithmic expression, using the properties of logarithms. We start with the expression \( \log_{5}(x) - 3 \).

Step 2: Apply the Power Rule of Logarithms

Recall that the power rule of logarithms states \( a \log_b(c) = \log_b(c^a) \). In this expression, the numeric term -3 can be interpreted using the power rule as a logarithm: \( -3 = \log_{5}(5^{-3}) \).

Step 3: Use the Subtraction Rule of Logarithms

The subtraction rule of logarithms states \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \). We will apply this rule to combine \( \log_{5}(x) \) and \( \log_{5}(5^{-3}) \).

Step 4: Combine into a Single Logarithm

Using the subtraction rule from the previous step, we have \( \log_{5}(x) - \log_{5}(5^{-3}) = \log_{5}\left(\frac{x}{5^{-3}}\right) \).

Step 5: Simplify the Expression

Simplify \( \frac{x}{5^{-3}} \) to get \( x \cdot 5^3 \), since \( \frac{1}{5^{-3}} = 5^3 \). Therefore, the expression becomes \( \log_{5}(x \cdot 5^3) \).

Step 6: Final Expression

The single logarithmic expression is \( \log_{5}(125x) \), since \( 5^3 = 125 \).

Key Concepts

Properties of Logarithms

Logarithms have numerous properties that make manipulating and simplifying expressions easier. The main properties are instrumental in combining and breaking apart logarithmic expressions.
Understanding these properties can help in a variety of mathematical problems, including solving equations and converting expressions. Here are some key properties:

• 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: \( \log_b(M^n) = n \cdot \log_b(M) \)

These rules help in transforming complex logarithmic expressions into simpler forms.

Single Logarithm Expression

Reducing an expression into a single logarithmic form can make it easier to handle, especially when dealing with multiple log terms.
The objective is to combine various logarithms into a unified representation.

Consider the expression \( \log_5(x) - 3 \). The task is to transform this into a single logarithmic term. We need to express the numeric term \(-3\) as a logarithm to make use of logarithmic rules. By interpreting \(-3\) as \(\log_5(5^{-3})\), it becomes possible to apply logarithmic rules to simplify the expression into a single term.
This method provides both clarity and convenience when solving logarithmic equations.

Power Rule of Logarithms

The power rule is an essential property of logarithms that is widely used to simplify expressions.
It states that \( a \cdot \log_b(c) = \log_b(c^a) \). This rule permits you to move a multiplier inside the logarithmic function as an exponent of the argument (c).

In the given exercise \( \log_{5}(x) - 3 \), the numeric \("-3"\) can be effectively turned into a log term \(\log_{5}(5^{-3})\) using the power rule.
This conversion allows the expression to be further simplified using the subtraction rule of logarithms.

Subtraction Rule of Logarithms

The subtraction rule, also known as the quotient rule of logarithms, is used when you have the difference of two logarithmic terms. This rule states that \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \). Essentially, it combines the two logarithms into a single expression by creating a fraction.

Applying this rule to \( \log_5(x) - \log_5(5^{-3}) \) results in a single logarithmic term \( \log_5\left(\frac{x}{5^{-3}}\right) \).
This fraction can be simplified to \( x \cdot 5^3 \), leading to the final simplified expression, \( \log_5(125x) \).

Mastering the subtraction rule simplifies many logarithmic operations.