Problem 35 - Algebra 2

Question

Use the properties of logarithms to evaluate each expression. \(\log _{3} 3+5 \log _{3} 3\)

Step-by-Step Solution

Answer

The value of the expression \(\log_{3} 3+5 \log_{3} 3\) is 6.

Step 1: Identify the Properties

Firstly, observe the properties that will be used. The logarithm base b of b is one, \(\log_{b}b = 1\). Then, the power rule of logarithms, \(\log_{b}a^c = c \log_{b}a\). Both of these will be used to simplify the expression.

Step 2: Apply the Properties

Next, replace \(\log_{3} 3\) using \(\log_{b}b = 1\) to get 1. So, \(\log_{3} 3+5 \log_{3}3 = 1+5(1)\).

Step 3: Simplify the Expression

Simplify the expression by performing the operations. So, the expression \(1+5(1)\) equals 6.

Key Concepts

Logarithm Basics

Logarithms play a crucial role in mathematics by allowing us to express exponential relationships in a more manageable form. Consider a logarithm as the inverse operation to exponentiation. If you know that an exponent represents repeated multiplication, a logarithm represents repeated division. More formally, for a given positive number base, such as 3, the logarithm tells us what power we need to raise this base to achieve another number.
For example, if you have \(\log_{3} 9 = 2\), it means that if you raise 3 to the power of 2, you get 9. Understanding this inverse relationship helps in simplifying expressions and solving equations involving exponents.

• The expression \(\log_{b} a\) asks: To what power must b be raised to get a?
• The basic identity: \(\log_{b} b = 1\) because any number to the power of itself is always 1.

Logarithmic Expressions

Logarithmic expressions often need simplification to be useful in equations or calculations. Simplifying logarithmic expressions makes them easier to manage and solve. These expressions often include terms like \(\log_{b} a\), and numbers can be multiplied or divided under the same log base.
For our specific problem, we handle the expression \(\log_{3} 3 + 5\log_{3} 3\). Begin by identifying terms that simplify independently. Thanks to understanding the basics of logarithms, we know \(\log_{3} 3\) simplifies to 1, given \(\log_{b} b = 1\). Each instance of \(\log_{3} 3\) in our expression represents '1'.

• Typically, you will use properties like the product, quotient, or power rules to simplify.
• Turn terms into a single form so further operations can be performed seamlessly.

Power Rule of Logarithms

The power rule is a powerful tool that simplifies logarithmic expressions involving exponents. This rule states that \(\log_{b} a^c = c \log_{b} a\). Intuitively, it lets us "bring down" the exponent as a multiplier, making logarithmic expressions simpler and easier to evaluate directly.
Returning to the exercise at hand, the power rule can simplify terms like \(5 \log_{3} 3\) by imagining it as \(\log_{3} (3^5)\). Although you could apply this method to break down the expression further, in our original expression, it's faster simply by multiplying since \(\log_{3} 3 = 1\). Thus, it directly becomes 5 times 1. Such rules adjust complex expressions into simpler arithmetic, allowing us to solve problems more effectively and quickly.

• Use the power rule to reconfigure terms where exponents are part of the logarithmic function.
• Speeds up evaluating expressions by direct multiplication when bases coincide.