Problem 32
Question
Expand each expression using the properties of logarithms. \(\log _{5} a^{-5}\)
Step-by-Step Solution
Verified Answer
The expanded expression is \(-5 \cdot \log_{5}(a)\).
1Step 1: Apply the Power Rule of Logarithms
The power rule of logarithms states that \(\log_{b}(x^n) = n \cdot \log_{b}(x)\). In this case, we have \(a^{-5}\) as our argument, so we can rewrite the expression \(\log_{5}(a^{-5})\) as \(-5 \cdot \log_{5}(a)\).
2Step 2: Simplify the Expression
The expression \(-5 \cdot \log_{5}(a)\) is already simplified. This represents the expanded form of the original logarithmic expression.
Key Concepts
Power Rule of LogarithmsLogarithmic ExpressionsExpanding Logarithms
Power Rule of Logarithms
The power rule of logarithms is a powerful tool in simplifying complex logarithmic expressions. When you encounter an expression of the form \( \log_{b}(x^n) \), you can simplify it using the power rule. This rule states that \( \log_{b}(x^n) = n \cdot \log_{b}(x) \).
This means that any exponent on the argument of a logarithm can be brought down as a coefficient outside the logarithm.
Let's consider the expression \( \log_{5}(a^{-5}) \). Here, \( a^{-5} \) indicates that 'a' is raised to the power of -5. According to the power rule, we can rewrite this as \( -5 \cdot \log_{5}(a) \).
It's important to remember this rule because it helps in breaking down and simplifying expressions, making them easier to understand and solve.
In schools, this concept is often used in contexts requiring simplification before further mathematical operations can be performed, such as in calculus or algebra.
This means that any exponent on the argument of a logarithm can be brought down as a coefficient outside the logarithm.
Let's consider the expression \( \log_{5}(a^{-5}) \). Here, \( a^{-5} \) indicates that 'a' is raised to the power of -5. According to the power rule, we can rewrite this as \( -5 \cdot \log_{5}(a) \).
It's important to remember this rule because it helps in breaking down and simplifying expressions, making them easier to understand and solve.
In schools, this concept is often used in contexts requiring simplification before further mathematical operations can be performed, such as in calculus or algebra.
Logarithmic Expressions
Logarithmic expressions involve logarithms, which are mathematical operations that are the inverses of exponentiation. You usually see them in equations where you have to find out the exponent that a base number is raised to achieve a certain value.
For example, the expression \( \log_{b}(x) \) signifies what power you need to raise 'b' to obtain 'x'. These expressions are common in calculations involving growth, decay, or anywhere exponential relationships exist.
In our exercise example, \( \log_{5}(a^{-5}) \), the logarithmic operation is helping us express how multiple powers interact, in this case converting an exponential equation into a more solvable form using logarithms.
Understanding logarithmic expressions is important because they are often necessary to solve real-life problems involving exponential growth like population growth, radioactive decay, and financial interest calculations.
For example, the expression \( \log_{b}(x) \) signifies what power you need to raise 'b' to obtain 'x'. These expressions are common in calculations involving growth, decay, or anywhere exponential relationships exist.
In our exercise example, \( \log_{5}(a^{-5}) \), the logarithmic operation is helping us express how multiple powers interact, in this case converting an exponential equation into a more solvable form using logarithms.
Understanding logarithmic expressions is important because they are often necessary to solve real-life problems involving exponential growth like population growth, radioactive decay, and financial interest calculations.
Expanding Logarithms
Expanding logarithms refers to the process of breaking down a complex logarithmic expression into simpler parts. This process utilizes various properties of logarithms, such as the power rule, product rule, and quotient rule.
By expanding a logarithmic expression, you can make solving the equation easier or prepare it for further calculations in algebraic processes.
In our original exercise with \( \log_{5}(a^{-5}) \), the expansion involves applying the power rule to convert it into \(-5 \cdot \log_{5}(a) \).
Such expansions are particularly useful because they strip down logarithmic expressions to simpler terms, allowing for a clearer manipulation of terms and making equations more accessible to solve.
Expanding logarithms is a fundamental skill in mathematics that applies in various scientific fields, including engineering, physics, and finance, where complex relationships demand simpler forms for accurate analysis and calculations.
By expanding a logarithmic expression, you can make solving the equation easier or prepare it for further calculations in algebraic processes.
In our original exercise with \( \log_{5}(a^{-5}) \), the expansion involves applying the power rule to convert it into \(-5 \cdot \log_{5}(a) \).
Such expansions are particularly useful because they strip down logarithmic expressions to simpler terms, allowing for a clearer manipulation of terms and making equations more accessible to solve.
Expanding logarithms is a fundamental skill in mathematics that applies in various scientific fields, including engineering, physics, and finance, where complex relationships demand simpler forms for accurate analysis and calculations.
Other exercises in this chapter
Problem 32
In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=-0.05729 $$
View solution Problem 32
In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ 8 \log _{\frac{1}{2}} 2 $$
View solution Problem 33
In \(33-44,\) if \(\ln 2=x\) and \(\ln 3=y,\) write each of the natural logs in terms of \(x\) and \(y\) $$ \ln 6 $$
View solution Problem 33
In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=-1.1544 $$
View solution