Problem 78 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. $$ \log 3+\log 5 $$

Step-by-Step Solution

Answer

The expression \(\log 3+\log 5\) can be written as a single logarithm as \(\log(15)\).

Step 1: Identify and apply the logarithm properties

One of the logarithm properties states that \(\log_b(m) + \log_b(n) = \log_b(mn)\). In this case, both logarithms have the same base, which is 10 (though it's not written). Apply the property to the given expression to combine the two logarithms into one.

Step 2: Calculate the product of the two numbers

Calculate the product 3*5 which gives the result 15.

Step 3: Write the expression as a single logarithm

Using the computed product, you can then write out the single logarithmic expression as \(\log(15)\).

Key Concepts

Understanding Logarithm Properties

Logarithms have specific properties that can be very useful when simplifying complex expressions. One key property is the product rule. This rule states that when you add two logarithms with the same base, you can combine them into a single logarithm of the product of their numbers. Formally, this means:

• \( \log_b(m) + \log_b(n) = \log_b(mn) \)

In simpler terms, if you're adding the logs of two numbers, you can multiply those two numbers and take the log of the result instead. It's crucial to ensure that the logarithms have the same base before applying this property.
In the example provided, \( \log 3 + \log 5 \), the base is commonly understood to be 10, as it is a "common logarithm" where the base is implied when not explicitly written. Applying the product property helps us to simplify this expression.

Combining Logarithms Effectively

Combining logarithms is all about using the properties of logarithms to simplify expressions. As seen in the exercise, the expression \( \log 3 + \log 5 \) can be combined using the product property.
To combine, simply multiply the numbers within the logarithms, giving you 3 multiplied by 5 which equals 15. This approach streamlines the expression, creating a simpler form that's easier to work with.
Logarithms can also be combined by using the quotient and power properties, though they weren't needed in this particular exercise. Understanding these basic principles allows you to tackle a wide range of logarithmic manipulations efficiently.

• Quotient Rule: \( \log_b(m) - \log_b(n) = \log_b\left( \frac{m}{n} \right) \)
• Power Rule: \( n\log_b(m) = \log_b(m^n) \)

Creating a Single Logarithm

Once you've combined logarithms by applying the correct properties, the next step is to rewrite them as a single, simpler logarithmic expression. This process not only simplifies calculations but also presents the expression in a more standardized form.
For example, after using the product property on \( \log 3 + \log 5 \), you end up with \( \log(15) \). This concise form makes further mathematical manipulation or interpretation much more straightforward.
A single logarithm expresses a compact summary of an initially intricate expression. This simplification is valuable, especially in solving equations or when integrating logarithmic expressions within more complex mathematical problems. Remember, the goal of rewriting expressions as single logarithms is to achieve simplicity and clarity.