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Write each equation in its equivalent logarithmic form. 2-4 = 1/16
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 13
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(e2/5)
Guida verificata passo dopo passo1
Recall the logarithmic identity that relates the natural logarithm and exponentials: \(\ln\left(e^x\right) = x\). This means the natural logarithm of \(e\) raised to any power simplifies directly to that power.
Identify the exponent in the given expression \(\ln\left(e^{2/5}\right)\). Here, the exponent is \(\frac{2}{5}\).
Apply the identity by rewriting the expression as \(\ln\left(e^{2/5}\right) = \frac{2}{5}\).
Since the expression simplifies directly to a number, no further expansion is needed.
Thus, the expanded form of \(\ln\left(e^{2/5}\right)\) is simply the exponent \(\frac{2}{5}\).
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Properties of Logarithms
Properties of logarithms include rules such as the product, quotient, and power rules that simplify logarithmic expressions. For example, the power rule states that ln(a^b) = b * ln(a), which is essential for expanding and simplifying expressions like ln(e^(2/5)).
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5:36
Change of Base Property
Natural Logarithm and the Number e
The natural logarithm, denoted ln, is the logarithm with base e, where e is approximately 2.718. A key property is that ln(e^x) = x, which allows direct simplification of expressions involving e raised to a power inside a logarithm.
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4:47The Number e
Evaluating Logarithmic Expressions Without a Calculator
Some logarithmic expressions can be simplified exactly using known properties and values, avoiding the need for a calculator. Recognizing forms like ln(e^x) helps evaluate expressions quickly and accurately by reducing them to their exponents.
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5:14Evaluate Logarithms
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