Problem 51
Question
Write logarithm without an exponent or a radical symbol. Then simplify, if possible. \(\ln y^{7}\)
Step-by-Step Solution
Verified Answer
The expression simplifies to \(7 \cdot \ln(y)\).
1Step 1: Apply the Logarithm Power Rule
The logarithm power rule states that \( ext{ln}(a^b) = b \cdot \text{ln}(a)\). In this problem, we have \(y\) raised to the power of 7. Applying the power rule, we rewrite the expression as \(7 \cdot \ln(y)\).
2Step 2: Simplification
Since \(7 \cdot \ln(y)\) cannot be simplified further without additional information about \(y\), the expression is already in its simplest form.
Key Concepts
Logarithm Power RuleSimplifying Logarithmic ExpressionsExponential and Logarithmic Forms
Logarithm Power Rule
The logarithm power rule is a powerful tool in simplifying expressions involving logarithms. It shows us how to transform expressions with exponents into a more manageable form by utilizing the properties of logarithms.
The power rule states that for any positive number \(a\) and real number \(b\), we have the equation:
For instance, in the exercise with \(\ln y^{7}\), the exponent 7 is moved in front to get \(7 \cdot \ln(y)\). This helps simplify expressions and solve logarithmic equations more easily. The power rule doesn't alter the underlying value but makes the computation more straightforward.
The power rule states that for any positive number \(a\) and real number \(b\), we have the equation:
- \( \ln(a^b) = b \cdot \ln(a) \)
For instance, in the exercise with \(\ln y^{7}\), the exponent 7 is moved in front to get \(7 \cdot \ln(y)\). This helps simplify expressions and solve logarithmic equations more easily. The power rule doesn't alter the underlying value but makes the computation more straightforward.
Simplifying Logarithmic Expressions
When working with logarithmic expressions, simplification is often a necessary step to make them easier to work with. Simplifying involves using various properties of logarithms, like the power rule, product rule, and quotient rule, to rewrite expressions in a simpler form.
In the example \( \ln(y^7) \), we applied the power rule to reach \( 7 \cdot \ln(y) \). After applying the logarithmic power rule or any similar step, we check whether further simplification is possible. However, simplification depends on whether you have more known variables or additional information.
Without specific values for \(y\), \( 7 \cdot \ln(y) \) is already as simplified as it can get. Often you may not have to go beyond the use of logarithmic properties like the power rule if there’s no further information provided to reduce it.
In the example \( \ln(y^7) \), we applied the power rule to reach \( 7 \cdot \ln(y) \). After applying the logarithmic power rule or any similar step, we check whether further simplification is possible. However, simplification depends on whether you have more known variables or additional information.
Without specific values for \(y\), \( 7 \cdot \ln(y) \) is already as simplified as it can get. Often you may not have to go beyond the use of logarithmic properties like the power rule if there’s no further information provided to reduce it.
Exponential and Logarithmic Forms
Understanding the connection between exponential and logarithmic forms is crucial, especially when transforming one into the other. It offers an ease in solving complex equations and breaking down logarithms into simpler forms. These two forms are inversely related.
Exponential form can be expressed as \( a^x = b \), while the logarithmic form of this is \( \log_a(b) = x \). In this context, 'a' is the base, 'b' is the result of the exponential function, and 'x' is the exponent.
This conversion is often needed when handling equations involving powers of variables. In our example, since we dealt with natural logarithms \(\ln\), it suggests a base of \(e\). Hence, whenever we encounter expressions like \(\ln(y^7)\), converting between exponential and logarithmic forms is done naturally by recognizing this relation. It helps streamline the work and ensures more intuitive simplifications.
Exponential form can be expressed as \( a^x = b \), while the logarithmic form of this is \( \log_a(b) = x \). In this context, 'a' is the base, 'b' is the result of the exponential function, and 'x' is the exponent.
This conversion is often needed when handling equations involving powers of variables. In our example, since we dealt with natural logarithms \(\ln\), it suggests a base of \(e\). Hence, whenever we encounter expressions like \(\ln(y^7)\), converting between exponential and logarithmic forms is done naturally by recognizing this relation. It helps streamline the work and ensures more intuitive simplifications.
Other exercises in this chapter
Problem 51
Solve for \(x\). See Example 3 . $$ \log _{x} 81=2 $$
View solution Problem 51
Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=2 x^{3}-3\)
View solution Problem 52
Solve each equation. See Example 7 . $$ \ln \left(x^{2}+4 x\right)=\ln \left(x^{2}+16\right) $$
View solution Problem 52
Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. $$ h(x)=x^{6}-100 $$
View solution