Problem 55
Question
In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 5 \ln x-2 \ln y $$
Step-by-Step Solution
Verified Answer
The condensed expression is \(\ln(\frac{x^5}{y^2})\).
1Step 1: Apply the Power Rule
The given expression is \(5 \ln x - 2 \ln y\). The power rule of logarithms allows us to move the coefficients inside the logarithm as an exponent of the argument. Apply this rule to rewrite the expression: \(\ln x^5 - \ln y^2\).
2Step 2: Apply the Quotient Rule
The expression is now in the form \(\ln(a) - \ln(b)\) which corresponds to the quotient rule of logarithms: \(\ln(\frac{a}{b})\). Use this rule to condense the two logarithms into a single logarithm: \(\ln(\frac{x^5}{y^2})\).
3Step 3: Check the Coefficient
Check that the coefficient in front of the logarithm is 1 (as required by the task). In this case, there is no coefficient other than 1, so we have satisfied the task requirement so no further action is necessary.
Key Concepts
Power Rule of LogarithmsQuotient Rule of LogarithmsCondensing Logarithmic ExpressionsNatural Logarithm
Power Rule of Logarithms
The power rule of logarithms is a tool that greatly simplifies the process of manipulating and solving logarithmic expressions. When you have a coefficient in front of a logarithm, you can 'move' this coefficient up to become the exponent of the argument in the logarithm.
To illustrate, if you have an expression like \(5 \times \text{log}(x)\), the power rule allows us to rewrite this as \(\text{log}(x^5)\). Similarly, the expression \(2 \times \text{log}(y)\) can be rewritten as \(\text{log}(y^2)\). This is useful in simplifying complex expressions and is a stepping stone for applying other logarithmic properties.
To illustrate, if you have an expression like \(5 \times \text{log}(x)\), the power rule allows us to rewrite this as \(\text{log}(x^5)\). Similarly, the expression \(2 \times \text{log}(y)\) can be rewritten as \(\text{log}(y^2)\). This is useful in simplifying complex expressions and is a stepping stone for applying other logarithmic properties.
Quotient Rule of Logarithms
The quotient rule of logarithms is equally essential. This rule addresses how to handle the division of two logarithmic expressions. When there is a subtraction between two logarithms, such as \(\text{log}(a) - \text{log}(b)\), the quotient rule lets us combine these into a single logarithmic expression by dividing the two arguments.
In other words, \(\text{log}(a) - \text{log}(b) = \text{log}(\frac{a}{b})\). This not only helps in condensing expressions but also is crucial when solving logarithmic equations where simplification is necessary.
In other words, \(\text{log}(a) - \text{log}(b) = \text{log}(\frac{a}{b})\). This not only helps in condensing expressions but also is crucial when solving logarithmic equations where simplification is necessary.
Condensing Logarithmic Expressions
Condensing logarithmic expressions is a process that makes use of various logarithm rules to rewrite multiple log terms into a single term. This is especially handy when you are trying to solve equations or simplify complex logarithmic expressions.
To condense an expression, you follow a sequence of applying the power, quotient, or product rule as needed. For example, given the expression \(5 \text{ln}(x) - 2 \text{ln}(y)\), you would apply the power rule followed by the quotient rule to condense it to \(\text{ln}(\frac{x^5}{y^2})\). This single logarithm is much easier to work with, whether for further simplification or solving for variables.
To condense an expression, you follow a sequence of applying the power, quotient, or product rule as needed. For example, given the expression \(5 \text{ln}(x) - 2 \text{ln}(y)\), you would apply the power rule followed by the quotient rule to condense it to \(\text{ln}(\frac{x^5}{y^2})\). This single logarithm is much easier to work with, whether for further simplification or solving for variables.
Natural Logarithm
The natural logarithm, denoted as \(\text{ln}(x)\), is a logarithm with base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. It plays a significant role in various fields including mathematics, physics, and engineering.
Properties of the natural logarithm are similar to logarithms of any other base; however, the natural logarithm simplifies many mathematical processes, particularly in calculus, due to its unique relationship with the exponential function \(e^x\). In the context of logarithmic properties, the natural logarithm follows the same power, quotient, and product rules as any other logarithm.
Properties of the natural logarithm are similar to logarithms of any other base; however, the natural logarithm simplifies many mathematical processes, particularly in calculus, due to its unique relationship with the exponential function \(e^x\). In the context of logarithmic properties, the natural logarithm follows the same power, quotient, and product rules as any other logarithm.
Other exercises in this chapter
Problem 55
Begin by graphing \(f(x)=\log _{2} x\). Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to de
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In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places.
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Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) Round answers to the nearest cent. S
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Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions.
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