Problem 15
Question
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{b} x^{3}\)
Step-by-Step Solution
Verified Answer
The expanded logarithmic expression is \(3 \cdot \log_b(x)\).
1Step 1: Applying Power Rule of Logarithms
By the power rule of logarithms, the rule is \( \log_b(x^n) = n \log_b(x) \). So, apply this rule to \( \log_b(x^3) \), replacing \( n \) with 3 to get an equivalent but expanded form: \( \log_b(x^3) = 3 \cdot \log_b(x) \).
2Step 2: Identify the relevant trigonometric identities
Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.
3Step 3: Apply the identities and simplify
Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.
4Step 4: Solve or evaluate
If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.
5Step 5: State the result
Express the final answer, including all solutions in the required domain if solving an equation.
6Step 6: Conclude with the answer
The expanded logarithmic expression is \(3 \cdot \log_b(x)\).
Key Concepts
Logarithmic ExpressionsPower Rule of LogarithmsExpanding LogarithmsEvaluating Logarithms
Logarithmic Expressions
Logarithmic expressions, at their core, are a way to represent the inverse operations of exponentiation. This means that if you have an exponential equation like \( b^y = x \), the logarithmic form would be \( \text{log}_b(x) = y \). The base \( b \) of the logarithm corresponds to the base of the exponential expression, and the argument of the logarithm \( x \) is the result of the base raised to the power of the output \( y \).
When working with logarithms, it's vital to understand their basic properties because they allow you to manipulate and simplify logarithmic expressions while solving or expanding them.
When working with logarithms, it's vital to understand their basic properties because they allow you to manipulate and simplify logarithmic expressions while solving or expanding them.
Power Rule of Logarithms
The power rule is a fundamental property that makes evaluating logarithms simpler. It states that a logarithm with an exponentiated argument can be rewritten by taking the exponent out front as a coefficient: \( \text{log}_b(x^n) = n \text{log}_b(x) \).
In practical terms, this property makes it easier to handle larger or more complex numbers, since dealing with exponents inside a logarithm directly can be challenging. It's especially helpful when you're trying to expand logarithmic expressions or break them down into more manageable parts.
In practical terms, this property makes it easier to handle larger or more complex numbers, since dealing with exponents inside a logarithm directly can be challenging. It's especially helpful when you're trying to expand logarithmic expressions or break them down into more manageable parts.
Expanding Logarithms
Expanding logarithms is the process of taking a single logarithmic expression and expressing it as the sum, difference, or product of simpler logarithms. This is possible thanks to various properties of logarithms, including the power rule, product rule (\( \text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y) \)), and the quotient rule (\( \text{log}_b(x/y) = \text{log}_b(x) - \text{log}_b(y) \)).
By expanding logarithms, you convert complex expressions into a form that is often easier to work with, especially when solving equations. Expanding also prepares the expressions for further operations, such as differentiation in calculus, or makes it easier to evaluate them without a calculator when the argument fits known logarithmic values.
By expanding logarithms, you convert complex expressions into a form that is often easier to work with, especially when solving equations. Expanding also prepares the expressions for further operations, such as differentiation in calculus, or makes it easier to evaluate them without a calculator when the argument fits known logarithmic values.
Evaluating Logarithms
Evaluating logarithms involves finding the value of logarithmic expressions. Sometimes, this can be done by recognizing patterns or using basic logarithmic values, such as \( \text{log}_b(1) \), which is always 0 regardless of the base \( b \), or \( \text{log}_b(b) \), which is always 1. Other times, you may need to apply logarithmic properties to simplify the expression before calculating its value.
In our exercise, evaluating \( \text{log}_b(x^3) \) using the power rule allows for a simpler representation which can then be more straightforward to evaluate, particularly if \( x \) is a value that produces a known logarithmic result. Making logarithms easier to handle not only aids in mental calculations but also lays the groundwork for more complex algebraic manipulations.
In our exercise, evaluating \( \text{log}_b(x^3) \) using the power rule allows for a simpler representation which can then be more straightforward to evaluate, particularly if \( x \) is a value that produces a known logarithmic result. Making logarithms easier to handle not only aids in mental calculations but also lays the groundwork for more complex algebraic manipulations.
Other exercises in this chapter
Problem 15
Write each equation in its equivalent logarithmic form. $$13^{2}=x$$
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Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{2}\right)^{x
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An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbon-I4 present after t years. Use
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Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$7^{\frac{x-2}{6}}=\sqrt{7}$$
View solution